
Walking Chocobo: First go to the chocobo tracks near the Golden Saucer, there you will find the "walking" breed of chocobo. Wander around on the tracks until you get into a battle with a chocobo in it. What you need is a battle that has two "Spencers" and a chocobo. You you get anything else other than two s"Spencers", just end the battle quickly and try again. Once you get your chocobo, go back to the stables and name it something that you can remember (be sure to have bought all 6 of the stables).
Running Chocobo: Now go to the tracks near Mideel to capture a Running Chocobo. Wander around on the tracks like before and wait to get into a battle with a chocobo and two "Spirals." Capture the chocobo. Now go back to the farm and make sure that the chocobo that you just caught is of the opposite sex of the Walking Chocobo. Be sure to name it something that you will easily remember. Currently, in your stalls you should have a walking chocobo and a running chocobo that are both of the opposite sex.
Before you breed be sure to save, as you'll see why later. Now go to the grassy field outside of Bone Village. Wander around until you get into a battle with a Vlakorados (big red raptor looking monster). He has a very high HP but his attacks are really really weak. Steal a Carob Nut from him and then slaughter him. Now go back to The chocobo farm and save your game. Now go and talk to Choco Billy and tell him that you want to breed your chocobos. He will ask you which ones that you want to breed and you gotta pick your two chocobos that you just got. Now he'll ask you what nut you want ot give them. Be sure to say that you want to give them the Carob Nut that you just got. Now they will breed. You should get a green chocobo or a blue chocobo if you raced you chocobos at the golden saucer like I told you to. If you actually do get a yellow chocobo, just reset it and try again. Depending on which chocobo you got, you should go to the Cave by Wutai to get the Mime materia if you got a green chocobo, or go to the cave on the tip of the island of Mideel, to get the Quadra Magic materia if you got a blue chocobo. Now, feed 40 Sylkis Greens to your chocobo to get him (make sure its a male) to the max, then race HIM up to S Class. (tip: do the item duplication trick to get 99 Sylkis Greens. Click here to find out how to do this.
Now, wait for about 20 minutes doing whatever (I usually get some items at the Battle Square). Also, in this 20 minutes, go get another Carob Nut. When 20 minutes is up, go back to the Chocobo Farm and breed your Walking Chocobo and your Running Chocobo again with another carob nut. Make sure that you get a chocobo that is not what you got before (if you got a green one last time, make sure you get a blue one this time and visa versa). Also, make sure that it is a female. Do the same process to the chocobo that you did above (40 Sylkis Greens, Racing up to S Class).
Now mess around for another 20 minutes and go get another carob nut again. When 20 minutes is up, go back to the farm and save. Now feed breed your blue and green chocobos and give them a carob nut. You should get a black chocobo (if not, try again). With this chocobo, you can access the cave by Corel. Inside is a HP-MP materia. It sux so don't even use it. Now race this one up to S Class...bla bla bla.
Dashing Chocobo: Now go up to the tracks on the Ice Continent and wander around until you get into a battle with a chocobo and two "Jumpings". Now get this chocobo that you caught and race it up to S Class and feed it 40 Sylkis Greens. Now go to the island with a little forest on it in the North East part of the map. Wander around in that forest until you fight a goblin dude. Steal a Zeio Nut from him and then slaughter him. Now go back to the farm and SAVE! Now go and breed this dashing chocobo with your black chocobo and you'll get a GOLD CHOCOBO! With this chocobo, go to the island in the very top right of the map. Inside of that cave you will find a Knight of the Round Summon Materia! Its the best summon in the game. Well, happy gaming!