Each Chocobo requires a breeding item as somewhat of a catalyst, and can then be bred with another Chocobo. Initially one will only have regular yellow Chocobos, and to change that the first thing that is required is a Great female Chocobo and a Good male Chocobo. Feed them a Carob Nut, and there will be a potential to obtain a Green Chocobo, which is able to travel across land and beaches like normal Chocobos, and also cross mountains.
Another thing to note is that the Nuts required to breed these Chocobos can only be found in specific places. The Carob Nut can be stolen by enemies outside of the Bone Village, and the Zeio Nut, which comes into play later on in the breeding process, can only be stolen from Goblins located on a small island known as Goblin Island.
After obtaining the Green Chocobo, one should try for a Blue Chocobo, which can cross small rivers but not mountains. In order to do this, a Great male Yellow Chocobo and a Good female Yellow Chocobo must be mated using another Carob Nut, and the potential to obtain a Blue Chocobo will be there.
After obtaining both a Blue and Green Chocobo, one can move on to creating a Black Chocobo. One of the colored Chocobos must be male while the other female. Once the Blue and Green are mated with a Carob Nut, it is possible to get a Black Chocobo, which can travel across land, small rivers, and mountains.
To get the ultimate Gold Chocobo, the Black Chocobo obtained must be mated with a Wonderful Yellow Chocobo of the opposite sex, using a Zeio Nut. There will then be a chance for the offspring to be a Gold Chocobo, which can essentially cross anything, including the ocean.
The point of this entire task is for a massive stock of rewards. With a Gold Chocobo, players are able to go to the waterfall cave without the submarine, and the Ancient Forest without defeating Ultimate Weapon. Not only that, but the player also will have access to four caves scattered across the world that would otherwise be inaccessible, each providing a powerful Materia that can be used to devastate opponents.