turn left while runnign/flying left,
turn right while running/flying right

speed increases (check with speed display, chatcommand: /speed )

on the ground, in air,
works everywhere if paced right.

The lower your speed is, the faster you have to turn.
The higher your speed is, the slower you have to turn
to get the same speed-boost from turning.

You need to learn a sense of proportion.

Start by running on the ground with pressed W and D or A and turning RIGHT or LEFT , appropriately to your pressed direction
and use speed display to see if your speed increases (type /speed into chat).
Your speed should increase, from 250 of straight run into 275 of run while turning.
This exact turn speed will help you increase your speed and sitance of any jump, flight, fall, everything.

Sometimes you can get only 170 speed and you need to have the speed boost from air-strafe to make the jump and you must run much faster then.

Watching other players which are good will help you see at what speed they turn how fast.

First, open Options and Keybindings and there go through every single option and set it for yourself. Change their binds to what is for you handy! Also pay attention to mouse sensitivity.
Most important(default):
Movement(WSAD) , Jump(space/mousewheel), Crouch/duck (ctrl), Save checkpoint , Load checkpoint , Join spectate mode , Change Spectate Camera , Respawn/Spawn , Switch weapon (q) .
You also may need to enable console, if by clicking ` ~ you dont open it.

Save Checkpoint on F5 and load checkpoint on F2 would be good. Also, keep Spawn and Join Spectate Mode on F10 F11 , just so they are away from Load checkpoint and you won't miclick them.

Weapons: USP is lightweight, Axe/Climbtool is heavy - lightweight is best for high speed, as the heavy will slow you down. You may need to Customize your Climber in Options of Main Menu to have USP available, then switch to it with (Q) LastWeapon or Select 1st or 2nd weapon buttons.

Mouse sensitivity is a long river to swim through. You need to be able to do 3 turns, each +120* angle in a second. If you cant' do it, increase mouse sensitivity.
Very good players move the mouse with their forearm on the whole desk, keeping mouse sensitivity low, so their movement can be precise.

Mousescrolldown for bunnyhop jump is already pushing it far in my opinion (why the ♥♥♥♥ make maps that are so damn precise you need to spam jump with scroll, if there can be and are maps for which jumping with Space is good enough? I have no idea).

Chat commands:
/speed - this will show speed in HUD (yours if you're playing, if you're spectating it will show
timeleft - displays the current time limit for the map
!top - this will display in a map-vote alike menu the top times of the map completion
!wr - this will display best achieved time of map completion from the community overall
if !top or !wr don't work, try top or wr, without !

Each time you make progress in map completion save the checkpoint. You can spam it. If you stand still, save the checkpoint! Click the save checkpoint button and go on.
You can of course save the checkpoints when you're in position for best attempt to the next jump - this will help save some time on pre-running.
Once you save checkpoint, you can load it anytime.

There is a separate time category for no-checkpoint runs but that's ahead of you, that's for later.

A good jump can be broken down into sections:
1. PRESTRAFE
2. JUMP
3. SNAKE
4. DUCKLAND
A very good jump does it all in 1 second total and flies on 260 units. That's future.

Sequence:
W+A/D+TURNLEFT/RIGHT > W+A/D+TURN+JUMP > A/D+TURN > D/A+TURN > A/D+TURN > DUCK
don't hold DUCK while turning, duck only to land

first click sidestep, then turn into it, then release it, click the opposite direction sidestep and turn into that.
You should get used to doing 3 zigzag turns with their sidesteps in 1 second. You can do it, jsut be quick.

Sequence on PrestrafeJumpRight:
W+D+TURNRIGT > W+D+TURN+JUMP > stop turning , click A and hold A and TURNLEFT > stop turning , click D and hold D and TURNRIGHT > stop turning, click A and hold A and TURNLEFT > TURNRIGHT > DUCK

doing W+D or W+A as the whole prestrafe works only if you turn consistently during the whole run before jump.
Type /speed to see the speed display in HUD and if you are running before jump on prestrafe and your speed is 270, you're doing good prestrafe.

Join spectate mode with the Spectate mode button from Keybinds, then switch Spectate Camera Mode form keybinds until you're in First-Person view, then use Mouse clicks to cycle through the players and watch who simply amazes you.

The green crosshair is very useful, as it shows what buttons the player is currently holding down / pushing / using.
The arrows represent Movement,
The J J represent Jump,
The D D represent Duck / Crouch.

Learn from the best! When to click what and how to navigate in the air. That will help you more than reading me. Also, feel free to talk to people - some of them really like helping others learn. Just be precise in asking what you want to learn (avoid "How you do this?!" - this goes nowhere. Also some players don't really like to try to explain, so better jsut watch them and see what when do they do).

Holding W will give you Max Speed of 250.
Holding W and pushing A or D with W will increase the speed to +270 while turning and this gives you the speed for longjump.

In air, release W - you don't need W in air, it's useless. Hold D or A down and turn - turning into direction you're stepping increases your speed.

Zigzag: MOUSE TURN LEFT WITH SIDESTEP LEFT, MOUSE TURN RIGHT WITH SIDESTEP RIGHT.
A+TURNLEFT > TURNRIGHT+D > TURNLEFT+A > TURNLEFT+D

How does it work? Source Engine uses PhysiX which calculate Speed from Momentum+Force. By turning and sidestepping, you add force to your player model, increasing the momentum, thus increasing the speed.

You should test youre proper turn speed:
display /speed, run in circle with W+A or W+D and turnig left/right, watching the speed count. You need to learn the proper turn speed, the one that increases your speed while running to 270. When you see it, keep turning at that speed and try to memorize it. It's muscle memory.
This turn speed is needed in all kinds of jumps/flights to increase your speed and distance of flight (longjumps whether high or low, surfing, falling, duck/crouchjumping, everywhere it's useful).

Very good climbers bunnyhop through climb maps doing the zigzag and prestrafe jumps all the time. I'm not there yet, but hopefully this guide helps you get there.
Good luck!

Lowjumps are on ground level of the map, by the floor and these you do on the last pixel point of the ramp, as far as you can walk - that's where you jump from (on prestrafe with W+A/D+TURN).

Highjumps are above the ground, at least a player-model height above the floor and these you do a little bit before that furthest point. If you highjump like a lowjump, you get slowed down. Why? That's the game engine.
13 units before the ramp finish is good place to jump on (with prestrafe). 13 units is like 1 centimeter on the monitor (just look down, see the edge of ramp, place your 5th finger on the monitor and your fingernail will connect the edge of ramp and the place you need to jump on/out from).