Instructions
Play through the game until reaching the second disc. During the first entire disc of the game you are unable to race chocobos, and instead will only be able to bet on the races.
Rent several chocobo stables from Choco Bill at the farm near the city of Kalm.
Equip the "Chocobo Lure" materia on one of your party members and catch a few chocobos. Choco Bill will evaluate their overall condition and let you know if they are worth keeping. Breed the chocobos found near Gold Saucer and the city of Mideel using a "Carob" nut item to produce higher level green and blue chocobos, which will fair much better in the races.
Breed a green and a blue chocobo together using a Carob nut to produce a black chocobo, which can run through all of the obstacles on the race, such as rivers and mountains, without slowing down at all. The black chocobo will also have more stamina and a higher running speed than the lower end chocobos.
Feed your chocobo several Sylkis greens, which can be bought from the Chocobo Sage north of Icicle Inn. The Sylkis greens will raise your chocobo's stamina and run speed even higher.
Go to Gold Saucer and enter the Chocobo Square. Speak with the person at the front and enter the races with your black chocobo.
Hold down the R1 and R2 buttons on the Playstation controller at the same time while racing to slowly refill your chocobo's stamina meter. This won't entirely prevent the chocobo from becoming exhausted during the race, but it will allow your bird to run much longer than normal before needing to slow down.
Press the "Circle" button on the controller when you are near the finish line to unleash one last burst of energy in your chocobo, dashing at a much higher speed than normal. This is especially useful if you are neck and neck with another chocobo racer near the end of the track.