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First off, look for claw prints, (I'm guessing claws since they are birds). If you hover around them, you can fight a battle against chocobos.
Now, I think its bad to hit them as they'll run off or die. I think you have to beat their accompanying enemies, to leave just the chocobo by itself, thus allowing you to capture it. However, the chocobo is easily scared, so you need nuts to feed the chocobo to keep it happy.
The nuts vary, and the more expensive the nut, the better it'll work, but that's how you capture them.
The breeding is a whole other matter though... and one I really can't remember properly. There are all sorts of classes of chocobo, and different types of nut to breed with. I hate to admit it, but a guide is very useful here if you want to breed blue, green, black and gold chocobos, rather than just more yellows.
One last bit of advice - gold saucer. Its an essential place to visit when building up chocobo stuff. I think its for getting S-ratings on your chocobos or something, but it might also be for winning nuts, I'm very hazy.
Anyway, good luck!