Here is a list of all mounts and how to get them:

Chocobo: Do mainstory line. And join Grand Company.
Legacy Chocobo: You must be an player from 1.0 that has legacy status.
Goobbue Mount: This was only avalible to get in version 1.0
Courel Mount: Buy CE or digital upgrade if ur a 1.0 player.
Magitek Armor: Do main story.
Unicorn Mount: Do all the conjurer class quest.
The flying eye(i dont remember the excact name :P ): get it as a Veteran Reward for paying 180 days.

i hoppe this helps!