To get a Normal Chocobo go to any Chocobo Tracks and use the materia "Chocobo Lure" and fight a chocobo. Feed it greens then kill the enemy's around it to catch it. If you get a chocobo in different areasd you may get better chocobos. I suggest the tracks by Mideel of to the North glacier town area. To get a Green chocobo (It can run across mountains) or a Blue Chocobo (It can run across shallow water) Breed a Good and Great Chocobo at the ranch using a Carob Nut. (for more info on this talk to Choco Billy in the stable. To get a Black Chocobo (It has the capabilities of a Blue and Green Chocobo) breed a Blue and Green Chocobo with a Carob Nut. You get carob Nuts at a small island south of Bone Village you steal it of a dinosaur like enemy called Vlakorados. Then to get a Gold Chocobo You breed a Black Chocobo with a Wonderful Chocobo with a pepio nut recieved at the eastern-most continent off of Goblins. You get Wonderful Chocos at the  north glacier town area. Although you dont always catch them.  
With a Gold Chocobo you can get the best Summon Knights of Round at a cave on a hidden island (it's not on the map) in the north eastern part of the world. I suggest trying to find it by airship first.

You can Use the Black Choco to get HP <--> MP (which swaps your Hp and MpTotals) at a cave by North Corel.

You can use the Green choco to get mime at a cave by Wutai.

And You Can use the Blue Choco to get Quadra Magic at a cave on the string of islands by Mideel.