You have to get a Chocobo license to be able to ride the chocobos. You will get that around lvl20 after you have joined a Grand Company(Its a sidequest that will be available to you in the city you choose)

For the Unicorn, you gotta get CNJ to lvl30 and unlock WHM. The Unicorn quest will then be available to you, and you just have to finish that to get your horise