For Chocobo breeding, you're gonna need to get the Highwind, (which you get somewhere in Disc 2), and a load of money. But luckily you fight quite a bit during the breeding process so can make money from that.
To catch a Chocobo, you need the Chocobo Lure from Choco Billy for 2,000 Gil. You should have bought it earlier in the game when you cross the marsh in Disc 1. If you sold it, you can get another one in the Chocobo Farm, by the outside pen, which should now be empty. It's to the right when you first go in, against the south side of the fence. Once you have a Chocobo Lure, equip it to someone and walk around on some Chocobo tracks until you enter a battle with a Chocobo. You can tell its a Chocobo battle due to the different music (sounds like surfer music!). DONT attack the Chocobo, just kill the enemies. You can feed it greens to distract it while you defeat the enemies.
Another way to catch Chocobo's courtesy of marioRPGfan.
To catch a Chocobo much, much easier, find your fastest person (The person who loads his time bar the fastest). Then, equip Odin materia to that person. As soon as you enter the battle, use Odin. If he uses Steel Bladed Sword, you're almost set. As long as it hits all of your enemies (Kills them all), you've got your Chocobo, because it ALWAYS misses the Chocobo. If he uses Gunge Lance, you don't have much of a chance of getting your Chocobo. This way, you shouldn't need greens. This will save you lots of money when catching Chocobo's.
Go to the Chocobo Farm and enter the house, not the barn and talk to the guy there to buy Chocobo stables for 10,000 Gil. If you are low on money, just buy three stables (but bare in mind you will need to buy at least one more stable later on).
Fly in the Highwind to the northern continent to the Chocobo Sage's house (the house surrounded by grass to the northwest of the farm. Fight in this area (on the grass) until you fight a red dragon/dinosaur type thing. You'll know what it is as I'm pretty sure the only other things you fight there are some things that look like squirrels. Have someone equipped with Steal and steal a Carob Nut from it. End the battle one way or the other and repeat until you have 3 Carob Nuts.
Get back in the ship and fly to the Gold Saucer. There are Chocobo tracks, catch a Chocobo here, (see above). Dismount the Chocobo and when the option appears select 'Send the Chocobo back to the stables'. Its a good idea to catch 4 Chocobos here and send them all back to the stables before returning to the Chocobo farm. Save before you enter.
Talk to Choco Billy in the stables and pick the 'Moving Chocobos' option. You will cycle through the Chocobo's you have in the fenced area in front of the stables. Choco Billy will rate the Chocobo's you have caught. You will need a 'good' Chocobo (they will be walking when you see them), not an average (which is standing still) or anything worse (with it's head down). Before moving the Chocobo into the stable, you will find out it's gender and name it. You will need a male one. If your only 'good' Chocobo is a female, you can reset from where you saved outside as gender is determined randomly. You should release all other Chocobo's from the pen when you have moved the 'good' one into a stable, by selecting 'Moving Chocobos' again, but instead of putting them in, choose to let them go.
If you have a 'good' male Chocobo from the area near Gold Saucer, fly to the Southern continent and land near where Cloud is found in Disc 2, Mideel. There are Chocobo tracks nearby. Catch another 4 Chocobo's here, see above on how to catch one, then fly back to the Chocobo farm. You can again save incase of wrong gender.
Again go the stable and choose to move a Chocobo into a stable. You will need a 'great' one this time, (they will be running when you see them) and the gender needs to be a female. Again release all Chocobo's from the pen when you have finished moving them.
You should now have two yellow Chocobo's in your stable. Go outside and save -very important to do so as you probably need to reset a few times before producing the right type of offspring. Return to the stables. Talk to Choco Billy and choose 'Mating Chocobos'. Go to the individual stables of your two Chocobo and 'talk' to them using O. When you have selected both, pick the top option, then choose to breed with the Carob Nut (which you stole from the red dino deally on the north continent). Then choose the top option to breed them.
You should have gotten a green Chocobo. If you got a blue or yellow Chocobo, you will need to reset and breed again. Refer to the above paragraph on how to breed. You will need to take note of your Chocobo's gender.
You won't be able to breed the yellow Chocobo's again for a while (come on, they're not machines!!) or breed the green Chocobo yet. So you can either wait for a while and get on with the storyline, or simply release your two yellow Chocobos (by choosing 'Releasing Chocobos' and selecting the two yellow ones), and catching two more from the same areas (see above). It doesn't matter which way you do it, but you will need to buy another stable for 10,000 gil. Again use the Carob Nut when breeding either your new yellow Chocobos or the existing ones. If you caught new ones. they will still need to be the 'good' male, and 'great' female.
Anyway, eventually, you should have a green and blue Chocobo, of different genders. You should have two stables free by the way, if not, release your yellow ones. In the Highwind, fly back to the area where you stole the Carob Nuts and enter the house. Talk to the guy here and choose to buy something. Buy either 20 or 40 of the Sylkis Greens (they cost 5000 Gil each) depending on how much money you have. (You will need 40 eventually, but can do with just 20 at the moment). If you are low on money, you can go to the area northeast of Mideel, on the Southern Continent. There is a mountain on an island, with two beaches on each side. On the beach to the right you can fight Sandworms that are quite easy to beat for 5000 Gil, allowing you to buy one Sylkis Green. It is a bit monotonous, but a good place to earn money, I think.
Go back to the Chocobo Farm and talk to Choco Billy again. Choose 'Feeding Chocobos' and give 10 Sylkis Greens to your green Chocobo, and 10 to you blue Chocobo, by changing the number of greens to feed them with the up and down button and then press O twice to feed the Chocobo's.
Go to North Corel. Use the ropeway to get to the Gold Saucer. Purchase a single ticket and go to the Chocobo Square. Talk to the girl (Ester?) who is standing in front of the Staffroom door. Choose to race, then choose either your blue or green Chocobo. Pick the top option twice and press Start at the Chocobo stats screen to begin the race. If you have either Tifa or Cid in your team, sometimes they will ask to ride the Chocobo instead of Cloud.
Your Chocobo starts off with a C ranking, you need to improve this to an A ranking by winning races. If you just let the Automatic Sequence race for you, you are very likely to win every race. If you choose to do manual control hold R1, R2 and square to go faster and not decrease stamina. It usually takes about 3 winning races to go up a ranking. Race your Chocobo's until they are both at an A ranking or higher.
NOTE: When your win a race, you will win an item. You can choose to take the item or the GP. I highly recommend taking the GP, as you will be racing a lot and will gain a lot of GP. You will very likely gain at least 300 GP while racing your green and blue Chocobos to A ranking. When you have 300 GP go the Wonder Square and exchange your GP for a Gold Ticket, which gives you an unlimited pass to the Gold Saucer. This is very useful as you will be returning here again to race other Chocobos, so allows your to save your money for greens.
When you are done and both Chocobo's are at an A ranking, leave the Gold Saucer and return to the Chocobo farm. Save outside. Go to the barn, talk to Choco Billy and choose to mate your green and blue Chocobo's together, using a Carob Nut. Refer above on the specifics. Hopefully, you will have bred a black Chocobo. If not reset and try again. Take note of your black Chocobo's gender.
Fly to the elongated island with a forest on it to the northeast of the Chocobo farm. If you have learned the Goblin Punch Enemy Skill, it is the same place you learned that. Wander around in the forest until you enter a battle with a Goblin (it has boxing gloves on). Steal a Zeio Nut from it, you only need one.
Now fly to the west side of the Northern Continent. There are Chocobo tracks in the snow near here. You will need to land the Highwind on the grass nearby. Catch 4 Chocobo's here. See at the top for how to catch a Chocobo. Return to the Chocobo farm. Save before entering the farm. Check your caught Chocobo's in the pen for a 'great' or 'wonderful' Chocobo. It must be the opposite gender to your black Chocobo. You will also need to buy another stable or release your green and blue Chocobo's. You may also want to save outside again, so you don't have to re-capture your yellow Chocobo.
Now you may need to return to the Chocobo Sage's house on the Northern Continent. If you bought 40 Sylkis Greens from him before you can skip this bit, but if you just bought 20, you will need to return to him to buy another 20 Sylkis Greens. If you are low on money go to where I told you before to make some money.
Return to Chocobo Farm. Feed 10 of the Sylkis Greens to your black Chocobo and 10 to your yellow Chocobo. Refer above on how to do this.
Now you need to return to Gold Saucer and race both your black and yellow Chocobos to A rank or higher. Refer above for racing tips. When they are at A rank or higher, return to Chocobo Farm. Again save outside before entering.
Go into the barn and speak to Choco Billy and choose to mate your black Chocobo with your new yellow Chocobo. Use the Zeio Nut. Wait until the next morning and hopefully you got a ................dum dum duuuum......GOLD Chocobo!!!!
This is where you wonder whether it was all worth it. Well let me tell you it is, for with your brand-spanking gorgeous Gold Chocobo, you can get the best Summon materia in the game, the Knights of the Round. To do so read on:
Talk to Choco Billy and pick the option 'Riding Chocobos' and pick your Gold Chocobo. Ride over to the Highwind, and press 'O' to get on with your chocobo. Park the Highwind on the Goblin Island (its the island to the northeast of the eastern continent), and ride the chocobo north-northeast from there to a mountain and forested island, that isn't on the map. Go into cave and grab the ultimate summon materia....Knights of the Round!!!!!
