How far into Heavensward before you get flying?

You have to unlock flight in each zone separately by completing certain side quests (and certain main scenario quests that take you there), and also seeking out 'aether currents' (which look like greenish swirling balls of air) and clicking on them which can be found scattered around each zone.

Use the aether compass in your key items list to triangulate the position of the aether currents (which you acquire automatically when you gain your first flying mount), as online lists might be inaccurate, due to SE changing the location of the aether currents in a recent patch (and also changing the quests needed to unlock them). Again, although this might not be accurate, you could try this page for more information.

A list of mounts including flying ones can be found here (which also mentions where and when you get them).

Good luck! :D