BREEDING
When you breed chocobos, you ideally want a chocobo 2 pedigrees higher then your current chocobo. This will give 100% chance to increase Pedigree. Using higher pedigree chocobos will give a better chance at higher stats and Tier 3 abilities, but will never increase your pedigree more then 1 per breeding. So, there is a lot of risk/reward to determine.
Stats are determined at random,
with 1-4 stats coming from the lower pedigree chocobo and the remainer coming from the higher pedigree. It is completely possible for your new chocobo to have lower stat ratings then the sire, but higher pedigree still means higher stats. Abilities and colors have a 50/50 chance of being passed on, your fledgling with have the color and ability of either of the parents, there is no 'mutating'.