This guide is written for the 0.3.x versions of the game; 0.4 reordered the tech tree and added some stuff, but the basic concepts in this guide should still apply.
I can't update this guide at the moment since the current game version won't run with Proton any more (at least on my machine). I'll post an updated version once I can run it and have gotten comfortable with the changes in the current updates.

Before thinking about what to put on your ships, you need to know what you want to use them for, and when you need them. Ships cost upkeep in Mission Control (MC), which is expensive, could be used for bases that give you resources and tech, and angers the aliens. If you aren't at war yet, only keep a fleet around if you want to use it soon. There are roughly the following uses, in this order:

• Shoot down an Alien ship to study
  
• Snipe an early Mothership or Assault Carrier (ideally after researching Exotics)
  
• Go after space assets of rival (human) factions
  
• Defend the orbits of Earth (and other planets you colonize)
  
• Intercept Alien fleets in space
  
• Destroy alien habs and bases

Do note that you can put most of these off for a very long time, as long as you manage hate levels and can defeat the Aliens when they land on Earth (which requires a few armies with a navy at 6.0 or higher, or a few nukes, by the early 30s). So you only ever need to build (very) early game and late game ships, allowing you to skip huge portions of the tech tree and the expense of building and maintaining a fleet for a large part of the game. If that's your game plan, you can skip pretty much the entire mid-game section. However, starting the fight earlier can be a fun challenge, and is basically a necessity on higher difficulties with their lower MC limit. Also, going to Jupiter early can be very worthwhile, but will likely lead to war with the Aliens in short order.

I will assume that you are mostly constrained by your reserves and production of Nobles, Fissiles and (in the mid-game) Exotics. In the beginning, it might seem like you have lots of those and MC is the hardest bottleneck, but this will change very quickly once you go to war and stop caring about Alien hate levels. Keep stockpiling them, and don't waste them in your ship designs (or stations; seriously, those Heavy Fusion Reactor Farms...). You can never have enough.
On the other hand, you will always have enough Water, Volatiles and Metals anyway, unless something went horribly, horribly wrong. So it's almost always a good idea to pay more of those in a ship design if it allows you to save on Nobles, Fissiles or Exotics while getting similar performance.
Even if you happen to get lucky with Nobles, you can always turn them into money (with Nanofactories or by selling antimatter from Supercolliders) to sink into fancy government Orgs and Direct Investment.

All ship designs need enough Delta-v to get where you need them, and enough acceleration to get there when they are needed. It's often a good idea to keep enough Delta-V for the return trip (which might have a less optimal launch window) so you need to at least double the Delta-V cost of your transfers. But don't add much more than that; Fuel is heavy and will kill your acceleration if you overdo it. With patrol ships, you only need to have enough Delta-v to catch the target, which depends on the difference in acceleration. As long as you're not very closely matched, you're unlikely to burn more than 10-20 kps of Delta-v. It's still advisable to keep a bit of a buffer, so you can maneuver in combat and return to your refueling station afterwards.

For acceleration, about 5 milligees are enough to get anywhere in an acceptable time, 10 milligees is better. If you want to intercept Alien ships, you need to exceed the acceleration of the slowest ship in their fleet, which can get close to 80 milligees for small ones. Most are around 30 milligees, and Motherships and older Assault Carriers are below 10 milligees. With most drives, 80 milligees of cruise acceleration will also give you very decent combat acceleration, so any more than that is overkill.
If you can't match the opponents' acceleration, you can also split your fleet and have both parts arrive within a few hours of each other. Since escaping prevents the enemy fleet from acting for a while, you can keep chasing them until they run out of Delta-v. Aliens are also likely to accept engagements if they are closely matched in combat strength, especially if you have more reinforcements nearby (but ships transporting Alien agents and fleets with Assault Carriers will usually try to escape anyway).
For attack ships you send after enemy stations, acceleration matters less, because the enemy will come to you (and you should make sure you can deal with the defenders before sending the fleet).

Don't waste research on drives that don't get you enough acceleration. A Thrust rating of 7.5 or more is usually fine, and anything above 5 might be serviceable if it doesn't have any glaring flaws. Below 5, mark the project as obsolete and immediately forget it exists. Yes, this includes anything electrothermal and electromagnetic. Grid, Lorentz and Helicon are the best of them, cost about 100k tech to get with all the prerequisites, and still fall just barely short of acceptable. Experienced players may be able to squeeze a few niche uses out of them, but unless you know exactly what you are doing and why, they are very likely a waste of research.

If you like a drive, make sure to research the better reactors for it as well. Especially for power-hungry drives (which many of the good ones are), reactor and radiator are a huge part of the weight, and better reactors are lighter and more efficient, sometimes by a huge margin.

• Cannons are just useless, never ever put them on anything. Leaving the slot empty to save mass is better.
  
• Missiles are great against anyone who neglects their Point Defense (PD) which includes the Aliens in the early game. Later on, not so much.
  
• Torpedos are too slow to be of any use in ship combat. Nuclear torpedoes have a niche use for a fleet with very high combat strength on paper to scare the AI into bad decisions (like expending their fuel running away, or defending places you have no intention of attacking). Antimatter torpedos use two slots and don't have twice as much combat value, so don't get them.
  
• Particle Beams are almost useless, since their range is so short. By the time you get close enough to use them, a laser or railgun would have finished the job already. Particle Beam PD isn't bad, though.
  
• Lasers are a great allround weapon, as long as the target doesn't have too much armor. They will chew through base defenses very quickly, but are completely useless against Motherships and Assault Carriers. The smaller ones also serve as PD when they don't have targets in range, while you want the larger ones against ships and stations. They have a somewhat weird dynamic in ship combat, where their target seems to take almost no damage for a long time, only to melt almost instantly once it gets close enough that the laser can burn through the armor. The Medium Batteries also go on the Layered Defense Arrays of your Bases and Stations, so make sure to research them whenever you can. All UV Lasers require Exotics, so you can put that Tech off until you've got a solid stockpile of them. Their increased range is pretty useful, especially for station and base defenses. If you struggle there, consider researching them and the Medium UV Phaser Battery, but don't put them on ships until you have plenty Exotics stocked up.
  
• Railguns are too slow to hit anything that moves unless you're right next to it (except for Motherships, Assault Carriers or ships out of fuel or with destroyed engines). They also have a hard time to get past serious PD, which drastically limits their uses beyond the early mid game. They do a lot of damage if they hit, are good for orbital bombardments, and may be able to distract PD enough to get some missiles through. Lots of small ones are best for that. The Medium Railgun Battery also gives you something to put on the kinetic slot of Layered Defense Arrays, and might occasionally score a hit on an Alien ship that gets too cocky. Don't count on it, though.
  
• Coilguns have fast enough projectiles to hit the occasional slower ship, and are otherwise just better railguns. Their projectiles are slightly lighter and do less damage on paper, but that is easily compensated by more of them hiting the target. Again, use the small ones on ships and the Medium Batteries for stations. The third Coilgun upgrade requires Exotics for a rather modest improvement, so you can safely put that off until late game. Heavier coilguns are however very useful against armored targets, if you have enough of them to get through PD (mixing them in with a lot of lighter ones can be very helpful here).
  
• Plasma is the best late game weapon against small and mid-sized ships (which the Aliens use A LOT), with the heavy nose-mounted ones being the most efficient. The hull-mounted Batteries aren't bad either, and the medium ones will finally put something useful on the kinetics slot of your Layered Defense Arrays.

You will have plenty of alternatives for armor, heat sinks, radiators and batteries, but most of them are strictly worse than choices you can unlock at the same time, or slightly later. Don't just go for the first one that unlocks, or you might waste research on something you will never use. Keep an eye out for choices that don't cost Nobles or Exotics (except for armor, you will usually find something comparable, or at least adequate, that doesn't), and look at how much you get for the weight of the component.
With armor, keep in mind that the cost is per ton, but better armor weighs less for the same protection. Adamantane Armor might look much more expensive than older armors, but effectively costs roughly the same. Since you can unlock it very early, and the only "better" armors waste require huge amounts of Exotics, it will be pretty much the only armor you'll ever put on ships (unless you completely run out of Nobles, and still need ships).
Also remember that using radiators with low vulnerability allows you to keep them extended during combat, which makes heat sinks obsolete, saving a utility slot, plenty of mass and some resources.
The list of actually useful components is pretty short, and all of them are guaranteed to unlock. Late-game ships will generally use Adamantane Armor, Tin Droplet Radiator, Quantum Battery and no heat sink. Lighter ships with low energy use might be better off with Superconducting Coil Batteries, but that's a pretty situational choice. If you don't want to wait for the Tin Droplet Radiator, you can get the Ionic Dust Radiator from Advanced Heat Management Concepts (required before going to Mercury) in between. Before that, the basic Aluminum Radiator is perfectly fine, possibly with a Lithium Heat Sink if you want to retract it and have a Reactor, Engine or Weapons that generate a lot of heat. If you want to go for energy weapons very early on, the Salt Water Battery will be the best option until you unlock the Quantum Battery (which you'll likely pick up early anyway, since it is on the way to the projects for the sixth councilor slot and for raising the MC cap).

• Gunships are mostly useless in the early or late game, but have a niche use in bombarding Alien bases during mid game, since putting a Green Phaser on the front gives you the best bombardment value per armor weight in the game that ignores point defense.
  
• Escorts are best tiny ship for most uses. They have two hull weapon slots (for missile boats) and two utility slots (for anything else). Also, two Escorts are often cheaper than a Monitor or a Frigate, and they have the lightest armor per weapon slot, which is great for heavily armored bombardment ships (when point defense isn't an issue).
  
• Corvettes are useless. By the time you get something worthwhile for the nose slot, you have engines that can move bigger ships.
  
• Frigates have four utility slots, and are light and cheap, which is great for early (or disposable) Marine transports and colony ships.
  
• Monitors have four hull weapon slots which makes for a great missile boat (you can even spare one for PD), and enough utility space for some extra magazines.
  
• Destroyers are the lightest ship that mounts a two-slot nose weapon, which is excellent for fast ships to hunt down Alien (or rival) fleets.
  
• Cruisers have six utility slots, which is by far the most cost effective for mid- and late-game utility ships that may need some of the utility slots to boost their drives. They can also have enough weapons for self-defense.
  
• Battlecruisers are the lightest ship that can use nose-mounted plasma, and the large nose-mounted laser has very decent range. Once you have engines that can move them, they are a great cost-effective all-round combat ship.
  
• Battleships can mount a large turret and have enough space left over for some PD, which makes them great if you want some ships in your fleet that can fire in all directions (to protect against flanking). Their armor is slightly heavier for the same protection than for Battlecruisers, and you'll want a bit more on the sides, so most designs will end up quite a bit slower and more expensive.
  
• Don't bother with anything larger unless you have very specific requirements. The armor on those hulls gets so heavy and expensive that you can usually have two or three lighter ships for the same cost. Spinal cannons are very good at cracking heavy armor, and super-heavy lasers can dismantle stations easily, but you don't have to do that very often, and smaller ships will also get the job done, while still being useful in between. In theory, they also allow you to concentrate more combat power on the battle, but Alien fleets are usually at least 3/4 light ships, so that isn't really necessary in practice.

This is what you use to shoot down your first Alien ship. There isn't much variety here, since cannons are useless, early armor is very heavy, and you don't have engines that allow you to send ships to other planets in reasonable time. All these designs are basically a tin can with a chemical rocket engine and as much missiles as you can fit on them. You don't really need any new techs here. A few Escorts or Monitors with Aluminum Radiators, Li-Ion Batteries and Krait Missiles are perfectly adequate for that. If it makes you feel better, slap on a point of Silicon Carbide Armor (the lightest of the starting ones) and maybe five at the nose so you don't instantly die to lasers. All you have at the start are chemical rockets, any of which has insane amounts of thrust, but nowhere near as much Delta-V as you'll want. So pick the most efficient one, which is the Venture Liquid Rocket. If you happen to have lots of Volatiles but little Water, use the Neutron instead. If someone researched Advanced Chemical Rocketry (it's a measly 1000 tech points), upgrade to the Nova Liquid Rocket. You can also research Magazines to add some more ammo (your ships will still go through their missile reserves disappointingly quickly). Getting better missiles might also help, but isn't strictly necessary. A dozen of those ships is also usually enough to kill a poorly protected Mothership - but make sure to research Exotics before you try, or those sweet, sweet resources will go to waste.
Since moving to the inclined Earth orbits (ISS and Tiangong) takes about 5 kps of Delta-v, you may want up to 10-12 kps on your early ships. This is also a bit above the amount of Delta-v Aliens can spend on escaping, given your fantastic cruise acceleration, which allows you to catch any Alien fleet you want to fight.

These ships are improvements on your first designs when you unlock better tech, and are still mostly missile boats built on the same principles. They are useful for harrassing other factions, escorting your Marines to their bases, and potentially for sniping a Mothership or Assault Carrier while the Aliens still neglect proper screening and PD. Make sure to research Exotics before hunting down a Mothership (Assault Carriers rarely leave you the choice). Even a single Mothership has enough Exotics to get you through the whole game if you use them smartly, but without the tech, they will go to waste, and the Aliens will be more careful next time. This is however completely optional, so you can just as well decide to lay low and quietly go up the tech tree. In any case, consider these ships disposable and only build them when needed. Keeping a standing fleet around wastes precious MC, and by the next time you need one, it will probably be obsolete anyway.
The most important difference is that you can now go for longer range engines, which is to say, fission. A ship at this stage should be able to easily travel to the Moon or between orbits, which requires about 10-20 kps. There is little need to travel to other planets; it's much easier to put a shipyard there and build your fleet in place. Having refueling depots at different orbits is also very useful (the Aliens love to use ISS and especially Tiangong orbit for landing, making control of those stations very useful).
There are many early game engines to choose from, but most are useless. The Pulsar is a decent engine you're guaranteed to get, and the Advanced Pulsar is a solid workhorse engine with a 40% unlock chance, that will stay adequate for a very long time. Heavy Dumbo is slightly better for heavy ships, and an alternative if you fail to unlock Advanced Pulsar. Advanced Cermet Nerva is a very expensive Plan C, and the rest of the Solid Core Fission engines are junk (and so is anything electric). If you miss out on Advanced Pulsar, going for Liquid Core Fission is generally the better bet. You are guaranteed to get Lars, which is better than any Solid Core Fission engine, and have a 50% chance to unlock Fission Spinner, which is better than any Gas Core Fission engine you can unlock for quite a while. Once you have Advanced Pulsar or Fission Spinner, you can safely abandon the Fission tree and focus on getting Fusion.
Otherwise, getting a Point Defense Laser to put on your ships might be a good investment, but don't bother researching any other ship upgrade until you get to the ones you want to keep using for a while. Notable in that regard are Quantum Battery, Ionic Dust Radiator, Liquid H2 Containment, Lithium Heat Sink (the best heat sink) and Adamantane Armor (which is the last armor tech you need). Avoid any component that uses Nobles like the plague, unless you don't have a decent alternative (like with Adamantane Armor).
Once you are set with these components, you can start to tech up the Laser and Kinetics tree (and into solidly mid-game ship designs). I prefer to go in the order of IR Laser, Visible Laser, Railguns, Arc Lasers, Coilguns, mostly research the projects for Green Arc Lasers and Small/Medium Coilguns and keep the other weapons for later.

These are ship designs that can easily travel to other planets, defend the inner Solar System and start to contest Alien fleets. Building them is entirely optional, you don't really need to fight at that stage of the game, and your limited MC is better used on more mining bases and research stations. But if you somehow managed to annoy the Aliens enough to find yourself at war, you may have little other choice. And if you prefer a very aggressive strategy, you might use these ships to colonize Jupiter early (which will likely end all your resource worries, but also get you to war with the Aliens very quickly).
If you lucked out earlier with the Fission Spinner drive, this is still perfectly adequate at this stage. Lars or Advanced Pulsar will start showing their age by now, so you might want to consider going for more advanced engines if you are expecting lots of fights soon.
Burner and Neutron Flux have good thrust and Delta-v and are guaranteed to unlock, but the former uses a bit of Nobles and Fissiles, and the latter consumes Fissiles at an alarming rate. The Orion engines have great thrust and good Delta-v, but use up quite a bit of Nobles and Fissiles and don't always unlock. Pegasus and Fission Lantern are good engines that only use up Water, but have a low chance to unlock.
If neither of those options appeals to you (or you don't feel the need to upgrade your fleet right now), this is a good time to start on Fusion. It requires quite a bit of up-front investment before you get anything useful, but you'll likely want it eventually, anyway. Also pick up Magnetic Nozzles while you're at it, since most good Fusion engines require it. Useful early Fusion engines are Triton Pulse (D-T-Fusion, Z-Pinch, Mag Nozzles), Advanced Helion Reflex (D-He3-Fusion, Mirror Cell, Mag Nozzles) and Triton Vista Inertial (D-T-Fusion, D-He3-Fusion, Inertial Plasma Confinement, Z-Pinch). The others are either too far up the tech tree or don't have enough thrust.
Slush Hydrogen Tankage and Muon or Neutronium Spikers are vital upgrades to get good range and thrust out of Fusion engines, so now is the time to get them.
To get the most from power-hungry Fusion engines, you should also upgrade to Tin Droplet Radiators (which you will use for the rest of the game). By now, you can also pick up Superconducting Coil Batteries, which are a good alternative to Quantum Batteries on lighter ships that aren't too energy-hungry.
At this stage of the game, putting 25 points of Adamantane Armor on the nose is enough to tank a few shots and protect you until you get into laser range. 5 points on the side and rear are usually enough to protect against (un-)lucky hits there. Your engines should be able to handle that amount of armor well enough, even on battlecruisers or battleships. Some players like to flip their battleships around and drift very slowly backwards when attacking stations. If you want to try this, you need to flip the armor as well (or armor both front and rear). I don't like it personally (and prefer using battlecruisers, anyway), but it is a viable design.
By now, researching Phasers and Plasma is also not too expensive (and you finally have engines to move your Plasma battlecruisers and batteries to power the Phaser banks). If you have the tech and Exotics to spare, you can also go for UV Lasers now, but there is no need to rush it.

These are the ship designs that can directly compete with (and eventually outclass) Alien ships. Once you have one you're happy with, build a dozen of them and go to war (if you haven't already). There is nothing more to gain by delaying. And then build a whole lot more of them.
At this stage, you should already have all the weapons and components you want, so the only thing left to get is a drive with enough thrust and Delta-v to force engagements with Alien fleets and fly around in the Kuiper Belt. This requires at least 500 kps of Delta-v (1000 if you want to be safe, more is excessive), about 30-40 milligees of acceleration to intercept most fleets in orbit and up to 80 milligees to hunt down smaller ships.
If you went for Fission engines before, you might be tempted to skip the entire Fusion tree and go for the Neutron Flux Torch and Antimatter Drives. As long as you have a decent supply of Fissiles, this is viable for a very fast and aggressive strategy. But getting Fusion also allows you to better deal with energy crises and slows down climate change a bit, which is vital for slower games, and generally safer. Even if you don't go for Antimatter engines, producing some antimatter and putting Antimatter Spikers on your ships adds a lot of thrust. Plus, you can sell the extra antimatter for insane amounts of money. Hydron Traps are also a very worthwhile use of Exotics on any ship that can equip them, and Exotic Spike Radiators may save you a bit of mass if you have the Exotics to afford them.
All Fusion drives you'll want at this stage require Terawatt Fusion, so get that now. Notable drives are:

• Zeta Helion (D-He3-Fusion, Z-Pinch), Mag Protium (H-H-Fusion, Tokamak) and Icarus Drive (D-D-Fusion, Hybrid Confinement), which are guaranteed, but cost a lot of Nobles to build, and have rather low thrust
  
• Icarus Torch (D-D-Fusion, Hybrid Confinement), which has decent thrust, but also costs a lot of Nobles and has only a 25% chance to unlock
  
• Zeta Boron (H-B-Fusion, Z-Pinch) and Protium Inertial (H-H-Fusion, Inertial Confinement), which are guaranteed and cheap, but also have low-ish thrust
  
• Daedalus Torch (D-T-Fusion, Inertial Confinement), which is cheap, efficient and has good thrust once you get the best reactor, but only has a 20% chance to unlock
  
• Protium Converter Torch (H-H-Fusion, Inertial Confinement), which is cheap to build (except for 2 Exotics a pop), has great thrust for a Fusion engine, but is very expensive to research and only has a 10% chance to unlock. It also requires so much power and cooling that anything else on the ship has negligible effect on acceleration or Delta-v, giving most reasonable ship designs about 100 milligees of acceleration (150 with Antimatter Spiker), at the cost of being almost unable to turn. Adding more engines won't make much difference on acceleration, but will ruin turn rate even more. Best used on heavily armored battleships or dreadnoughts (Steel or Silicon Carbide is best, since mass doesn't matter, but cost will).

Do not bother with the Firefly Torch. It looks good on paper, but requires way too much power and radiators, making your ships heavy, slow and almost unable to turn, without having the thrust to actually move them.