You're going to need a lot of money, patience, and luck if you want to raise chocobos.  The payoff is that you can breed chocobos that can fly, swim, climb over mountains and go to places normal vehicles can't reach.

    Here are some brief instructions.  Remember, there is a random element to Chocobo breeding, so it may take a couple of tries to get the kind of Chocobo you desire.  Also, be sure to save after correctly completing a step as you'll undoubtedly have to reset a couple of times during the breeding process.

    -After getting the Highwind in Disc 2, return to the Chocobo Barn.

    -You can purchase stalls from Choco Bill for 10000 gil per stall, and you'll need at least four stalls (you can purchase six total).  Remember, after you don't need chocobos any more, you should just kick them out to make room.

    - Before you can start breeding, though, you need to stock up on supplies.  Fly to either Bone Village or the Chocobo Sage's house.  If you run around on the grass, you'll eventually fight an enemy called Vrakoladus.  Fight that enemy three times, stealing a Carob Nut from it each time (you'll need the Steal materia first).  Then, go to the Goblin Island or Round Island and steal a Zeio Nut from the Goblins that live there.  Finally, go to the Chocobo Sage's house and purchase 40 Sylkis greens from him.

    - Equip a Chocobo Lure materia.  Fly to the chocobo tracks near the Gold Saucer (by the shore, check this map), and capture a chocobo.  Press X to dismount from the chocobo, and choose the option to return it to the Chocobo Barn.  You need to capture 2 chocobos from here, although you can get the other chocobo later or not get it at all, if you wish.

    - Now, fly to the chocobo tracks near Mideel.  Capture a chocobo here, and send it back to the Chocobo Barn.

IMPORTANT NOTE:  The gender of a Chocobo is not determined until you return to the Chocobo Barn and name them.  Therefore, it is advised that you save your game before naming them.  That way, if you don't get the kinds you want (see below), then you can reset and try naming them again and seeing if their gender changed.

    - Return to the Chocobo Barn and talk to Choco Billy.  Name your Chocobos (this will put them in the stalls).  Now, go up to your chocobos and examine them by pressing O.  You'll want to make sure that you have one male and one female (or two males and one of any gender, if you caught three chocobos).  Furthermore, one of them should be the running kind (usually you'll want the females), and the other should be the walking kind (usually, this should be the males).  Your third chocobo needs to be a 'runner' type.

    - So, you have at least one male and female, and one is a walker and the other is a runner, right?  If not, you'll have to release the chocobos that aren't the ones you want and go and capture more (or reset, if you heeded the important note).

    - Choose to mate the chocobos, and tell Choco Billy that you want to feed them a Carob Nut as well.  Hopefully, you'll get a colored chocobo the next day!  It should be green or blue.  If not, you'll have to reset and give it another go.  Keep note of your baby chocobo's gender--it's very crucial later on.

IMPORTANT NOTE 2:  Later on in this process, it's required that you race chocobos at the Gold Saucer (see below).  However, if you have trouble getting the correct type of chocobo during these few steps, you can optionally choose to race them.  The better your chocobo's race class is, the higher chance you have of getting the needed color type of chocobo.

    - At this point, waste some time, then mate your unused 'runner' chocobo with the green chocobo.  You can repeat the last few steps (get a chocobo from the Gold Saucer, get a chocobo from Mideel, make sure they're both the correct types, and breed them using a Carob Nut).  Or, if you got two other chocobos the first time around, you can then breed them together.  Your third choice is to use the same pair of chocobos that you just breeded.  In that case, you'll have to waste some game time, as chocobos that have just mated can't do it immediately again.

    - In any case, you'll want to breed your chocobos together and get a blue chocobo, using the steps outlined above.  Your blue chocobo MUST be the opposite gender of your green chocobo, or you can't mate them.  Feed the blue chick and the green chick 10 Sylkis Greens each.

    - Get in the Highwind and go to the Gold Saucer via North Corel's ropeway.  Go to the Chocobo Square and talk to Ester.  Tell her that you want to race using either your green or blue chocobo. The goal here is to keep racing until your chocobo's race level reaches A.  To do this, win the races consistently; it will take a couple of tries.  To insure winning, just use Automatic Mode for most of the race and hold R1 + R2 to recover lost Stamina while racing.  After one chocobo has hit rank A, do the whole racing process again with the other one.

    - Once your green and blue chocobos are both rank A, return to the Chocobo Barn.  Breed the two chocobos together, using your last Carob Nut (refer to the above steps for help).  If everything worked out, you'll have a black chocobo.

    - Fly to the Northern Continent (the snowy one), and look for Chocobo tracks on the western edge of the continent.  Land and then capture a chocobo at the tracks.  Return to the Chocobo Barn and name it (it should be the opposite gender of your black chocobo, and should be the 'dashing' kind--it runs very fast with it's head down).

    - Now for the fun part.  Feed 10 Sylkis Greens to the black chocobo and the dashing yellow chocobo.  Then, go back to the Gold Saucer and race both Chocobos up to race class A again!  When they're both at rank A, go back to the Chocobo Barn and tell Choco Billy that you want to mate them, using the Zeio Nut this time around.  With a little luck, you'll get the Gold Chocobo!

    Unlike wild chocobos that you can catch, the ones that you've named are tamer, and if you get off them, they will stay where you left them until you get back on.  You can also ride a tame chocobo into the Highwind and transport them around with you on the airship.

YELLOW CHOCOBOS can go anywhere you can normally get to on foot.  The sole advantage is that a chocobo is a lot quicker than walking.

BLUE CHOCOBOS can walk along rivers, but they can't go past waterfalls.  They can also walk in shallow water, just like Cid's airplane could.

GREEN CHOCOBOS can run over mountains, even the pointed mountains by Nivelheim.  They can also dash over _most_ cliffs and drops.

BLACK CHOCOBOS can walk along rivers and move around in shallow water.  They can also run over mountains and cliffs.  Unlike the blue or green chocobos, the black variety can go over any cliff or drop: even walk up or down waterfalls, enter the canyon surrounding the Ancient City, or walk over the ice forest that you have to traverse in Disc 2 after the snowboard scene!

GOLD CHOCOBOS can do everything a black chocobo can--they can also walk in the ocean, allowing you to go anywhere you want, even into the desert surrounding the Gold Saucer (just like the Saucer vehicle).  Truly the ultimate in transportation!

    In case you didn't know about it, you can rotate the chocobo you're viewing when using the various options in the Chocobo Farm by holding the R1 or L1 buttons :)