This page explains the necessary steps and requirements to unlock flying when mounted in Final Fantasy XIV.
How to Fly
If you already have flying unlocked, just jump while mounted to start flying. You can change it to require a double-jump to fly, by going to Main Menu > Character Configuration > Control Settings > General Tab, under "Flying Mount Takeoff".
How to Unlock Flying in A Realm Reborn Areas
To unlock flying in A Realm Reborn areas, you need to complete the level 50 main scenario quest The Ultimate Weapon. After you complete it, flying will automatically be unlocked in La Noscea, Thanalan, the Black Shroud, Coerthas Central Highlands, and Mor Dhona.
How to Unlock Flying in Heavensward Areas
You can begin to unlock flying in Heavensward areas after you complete the level 50 main scenario quest Divine Intervention. You will unlock an Aether Compass (accessible from Main Menu > Duty > Collection), which you can use to detect the nearest Aether Current. There will also be blue quests that unlock some of the Aether Currents. If there is no Aether Compass in your Collection window, read How to Get an Aether Compass to get one.
How to Unlock Flying in Later Areas
After you have completed Heavensward, you can use your Aether Compass (in Main Menu > Duty > Collection) to detect Aether Currents in the areas of subsequent game expansions. If you don't have an Aether Compass in your Collection window, read How to Get an Aether Compass. There will also be blue quests that unlock some of the Aether Currents.
How to Get an Aether Compass
You will get an Aether Compass during the level 50 main scenario quest Divine Intervention. It will be added to Main Menu > Duty > Collection. If you used a story skip and don't have an Aether Compass in your Collection window, you may be able to unlock it by talking to Gibrillont in the Forgotten Knight in Ishgard, or to Gaufrid in northeastern Rhalgr's Reach, or to Jesswren at the Amaro Launch in the Crystarium.