Basically, you'll only ever be able to unlock flying in zones where you've largely completed all story requirements. ARR being slight exception, you'll always be spending some time somewhere around sooner or later, but still requires completing 2.0. Starting with 3.0, where flying was originally introduced, the aether currents necessary to unlock flight are typically gated behind one final dungeon or trial in each zone. But again, only relevant in 3.0 and beyond.

Flying over obstacles is great, but for speed, you do get ground speed increases as well. Those will also come as story quests progress. They help slightly, with a second speed increase available for purchase with Allied Seals, tho I don't think those are very readily available at lower levels. And anyway, maybe no longer a wise purchase since, by the time you'll be able to start collecting them, flight makes it obsolete anyway.

Conclusion, all mounts are now flying mounts since sometime last year or something. But story quests will still and always gate when and where you can fly. Always progress MSQ. Additional small clarifications: no, there are no mounts that fly faster, they're all the same. And you will not have to find aether currents in ARR zones, they don't exist since flight was introduced long afterwards.