The Chocobo Mount is actually received from completing the quest through a Grand Company which I believe is lvl 20. Then you'll need to get roughly 2000 GC Seals by participating in Fates, Leve's, and provision quests to ear those seals. Then you can purchase what you need to complete the process.

Once you do all that you'll go into the Actions Tab on the bottom of your screen. Go to mounts, and then you can drag the chocobo icon into one of your hotbars.