There really isn't any "flying" mount, you will unlock the ability to fly (in the a realm reborn zones) by reaching level 50+ and completing the original mainstory, from that point all your mounts will be allowed to fly, no extra steps needed.

Unlocking flying in Heavensward aswell as the other expansions will than happen on a per zone basis, where you will have to collect aether currents/complete some quests, but the game will tell you about this in due time.

Edit: Blubb beat me to the punch