I discovered an easy way to catch birds!

First, you need a Torch (sold at John's once he's a resident).

Now, wait until night time.

Find a bird and flash the torch at it to stun it! If it worked, you'll hear a flash, and the bird will get stars circling over its head and (hopefully) fall to the ground.

Quickly switch to the cage (you only have a few seconds, so put them right next to each other in your tool belt), and cage the bird.

Pick up the bird to do with as you please.

Tada!