There are many different greens that you can feed your chocobos to strengthen their stats. You can buy these greens at the Chocobo ranch, but better and more useful greens can be bought at the Chocobo Sage's house on the Northern Continent. The Sylkis green, the one required to breed different colour Chocobos is found there as well.
When you want to breed two chocobos together, you will need "nuts". The 2 most important kinds of nuts are "Carob" and "Zeio" nuts. These nuts are stolen from different enemies, but you don't need large quantities of them unlike greens.