Instructions
Play through the main storyline of the game until you have possession of the Highwind. Without it, you will not be able to visit all the locations you need to in order to breed Chocobos.
Visit the Chocobo stables. You should have found the stables on the first continent at the beginning of the game. Talk to Billy in the stables and rent the maximum number of pens possible. Buy several greens from Billy as well.
Travel to the Chocobo tracks near the Gold Saucer. Send several Chocobos back to the stables. Fly back, and talk to Billy. Keep a Chocobo that Billy describes as "Good."
Head south to Mideel Island now and land near the Chocobo tracks. Repeat the process and catch several Chocobos. Fly back to the stables. You need a "Great" Chocobo of the opposing gender from the "Good" Chocobo that you caught earlier. Feed both Chocobos in the mating pair as many greens as you can afford to.
Go to the Gold saucer. Race the two Chocobos in your mating pair until you get them to the highest level possible. This increases the odds of getting the type of Chocobo you want when you breed them. Also while in the Gold Saucer, if you win ample amounts of GP buy several carob nuts at Wonder Square.
Return to the stables and talk to Billy. Save your game. Tell him you want to mate your breeding pair and feed them a carob nut to do so. You should get a blue or a green Chocobo. If you do not, reset the PlayStation and try again. You can also complete other portions of the game while you wait for the pair to be ready to breed again, but this can take quite a bit of time.
Repeat the process when the Chocobos are ready to breed again. Reset your PlayStation until you get a Chocobo of both the opposite color and gender. This can take many tries before it happens.
Feed and race the colored breeding pair of Chocobos until you have moved up at least a level or two in the rankings.
Breed the two colored Chocobos until you get a black Chocobo.
Fly up to the Icecle Inn. Find the tracks and capture Chocobos until you get a "Wonderful" Chocobo that is the opposite gender of your black Chocobo. Feed and level these two animals following the same process as used for the other Chocobos.
Travel to a long skinny island that is located to the northeast of the stables. Equip the "Steal" materia on someone. Walk around getting in random encounters in the forest until a Goblin enters the fight. Steal from Goblins until you get a Zeio nut.
Mate the wonderful Chocobo with a black Chocobo using Zeio nut. The result will be a Gold Chocobo.