If you've gotten to the level 20 story quests for choosing a Grand Company, after joining up and getting enough seals (I also believe you need to accept a quest after joining), you then can buy a mount license at the HQ. So for the Adders, it's in New Gridania; for the Flames it's right next to the atheryte plaza in Ul'dah, and for the Maelstrom it's right next to the Aftcastle, in the Limsa Lominsa Upper Decks.

Have fun in Eorzea