Hey Myraii, first, welcome to the forum, and welcome to the game!

Now, as you know to get your chocobo, you had to spend up those GC (Grand Company) Seals. For those different armors for your chocobo (known as 'barding') you've got a few options that I know of, there may be more.

1) You can buy another set of barding from your Grand Company for seals, i haven't actually done this myself though, so i have no idea what it looks like!
2) The collectors edition of the game comes with a new set of Behemoth Chocobo Barding, that's the ones with the purple horns type thing.
3) Being subscribed eventually rewards a black mage barding (3 months if i remember rightly, possibly wrong though)
4) And the other way i know is that so many player commendations gets you a chocobo barding. This can be checked in the achievements menu in game.