Ok- breeding a golden chocobo requires a lot of time and even more gil.
Breeding chocobos begins by capturing a female and male yellow chocobo. To catch a chocobo, equip the Chocobo Lure materia to a character and walk around areas of the world map that contain tracks. There is a 90% chance you'll enter a battle with monsters where a chocobo is present. Chocobos cannot be killed in battle, even with Omnislash. To catch a chocobo, defeat all enemies in the battle. After the battle is over, you will mount the chocobo. Take the chocobo to the Chocobo Farm and keep it in a stable. Stables cost 10,000 gil a piece.
Some chocobos require an incentive in order to stay on the battlefield. Use greens on chocobos to keep them from running away or attacking your party members.
The success rate of breeding a new colored chocobo depend on the class of chocobos. Winning races at the Gold Saucer will improve your chocobo. Classes start from C (lowest) to A (highest), then S (special). The higher the status, the better the chocobo. You can improve chocobos for later breeding by feeding them greens. You can buy greens from both Chocobo Billy and the Chocobo Sage. Greens improve chocobos in three areas: intelligence, stamina, and speed. The higher these values, the better breed your newborn chocobo will be.
Yellow Chocobo Resident Location(s) Chocobo Ranch Area (Poor & Weak)
Junon Area (Poor & Fair)
Gold Saucer Area (Average & Good)
Rocket Town Area (So-So & Great)
Wutai Area (Average & Fair)
Mideel Area (Great & Fair)
Icicle Area (Weak & Wonderful)
Traversing Ability This chocobo can traverse on land and beaches only.
Green Chocobo Breed via-Great Yellow Chocobo (Female)
Good Yellow Chocobo (Male)
Breeding ComponentsCarob Nut
Traversing AbilityThis chocobo can traverse land and over mountains.
Blue Chocobo Breed viaGreat Yellow Chocobo (M)
Good Yellow Chocobo (F)
Breeding ComponentsCarob Nut
Traversing Ability This chocobo can traverse land and rivers. It cannot traverse over deep water.
Black Chocobo Breed viaBlue Chocobo
Green Chocobo
Breeding ComponentsCarob Nut
Class A
Traversing AbilityThis chocobo can traverse land, rivers, and mountains. It cannot traverse over deep water.
Gold Chocobo Breed viaBlack Chocobo
Wonderful Yellow Chocobo
Breeding ComponentsZeio Nut
Stam 999
Spd 120+
Class A
Traversing AbilityThis chocobo can traverse any terrain and water.

If you want more info, go here-
Breeding Chocobos