Mount Armor?

[Nyalia]

Only your Company Chocobo is customizable. You can name them, dye their feathers, equip them with (vanity-only) barding (three slots: head/body/feet), level them up, and give them skills (three linear skill lines: tank/healer/DPS). You can also feed them treats which up one of their stats for a while, and if you train them (give them exp while stabled in your Free Company's stables) with one of the foods, it will become their favored food and you get a bigger bonus when using it.

When the Gold Saucer is released, you'll be able to race your chocobo, but the exact mechanics of that are not known.

When you hit level 20, after you join one of the Grand Companies in the main storyline, there is a quest at your Grand Company HQ that unlocks your Grand Company Chocobo ("My Little Chocobo"). Later, there is another quest at Camp Tranquil at level 30 that unlocks the ability to summon your chocobo to fight with you ("My Feisty Little Chocobo"). Note that the chocobo counts as a party member, takes a party slot when summoned, and reduces the exp you gain from combat by 25% (the choco gains twice the exp you miss out on).