The achievements needed are divided into two parts, one part which you can work on right away, and one part which requires you to play in patch 7.2 to unlock. The information from the second part of the guide might change once 7.2 hits live, because the data is only from the PTR as of now.
Remember to check back once patch 7.2 hits the PTR and the live servers for an updated guide!
The achievement you need to get to complete first part of the flying is called "Broken Isles Pathfinder, Part One". This is a meta-achievement, which means you need several other achievements in order to get this. The following requirements are needed to unlock the achievement:
1) Explore all of The Broken Isles.
2) Complete the major story lines in all Broken Isles zones.
3) Complete the class Order Campaign.
4) Complete 100 different World Quests.
5) Reach revered reputation with all Legion factions.
The requirements themselves are not too difficult, but require a serious time investment. The meta-achievement is account-bound, so you can do some steps on one character and other on another, and still get the achievement. This means that you are not tied to doing it all on your main.
Below you can read up on the details for each step of the achievement.
Broken Isles Explorer / explore all of Legion's zonesThis achievement requires you to explore all of the Legion zones completely, and it is actually a bit easier than the previous explorer achievements. It's pretty much just to go to any spot on the map you have not yet visited, and upon exploring the whole zone, the achievement will tick.
There are no especially hard-to-reach areas in the Legion version of this achievement, so go explore, you you will get it.
Note that this achievement is account-bound, so you can explore one zone on one character and the rest on another, and still get the achievement for exploring it all.
Loremaster of Legion / Complete all major quest chainsIn order to unlock this you need to complete all the major quest chains in all the Legion zones. Each individual zone has it's own achievement, and get all five of these to get the Loremaster achievement.
The four leveling zones should mostly come organically upon leveling up for the first time, but return to any quest chain you might have missed and finish it up to get it done.
The Suramar part can be pretty difficult, and actually requires a very lengthy quest chain, and most of it is gated behind reputation levels. You need to be exalted to complete the entire quest chain, so you better get working on the reputation to get this.
Variety is the Spice of Life / 100 different World QuestsThis part require you to complete 100 different World Quests, and this part will also come pretty much automatically for those of you that are into doing World Quests every day. Note that it has to be 100 different World Quests, so that doing some of them several times will not count.
If you have not yet unlocked World Quests, then quest in all zones until you are at least friendly with each faction. After doing this you can return to Khadgar in Dalaran to unlock them.
This part of the achievement also ties up with the next part, the reputation gains, because World Quests are the main source of rep gain in legion.
Broken Isles Diplomat / reach revered with all main Legion factionsYou need to be revered reputation with all six of the major Legion factions to get this part of the achievement, and the main source is as noted above: doing World Quests. You can read more about the different factions below.
1) Court of Farondis. These are located in Azsuna, and tied with the quests there. Completing all quests gets you to Honored level, and the rest of the grind comes from World Quests and Emissary Quests. Read the full Court of Farondis reputation guide here.
2) Highmountain Tribe. This reputation is tied with the zone called Highmountain, and you will get to Honored by questing here. The rest of the grind comes from World Quests and Emissary Quests. Click here for the full Highmountain Tribe reputation guide.
3) The Valarjar.You get reputation with these guys to Honored by doing quests in Stormheim. More rep is gotten from World Quests and Emissary Quests. Read the full Valarjar rep guide here.
4) The Dreamweavers. These guys are located in Val'sharah, and you get rep up to Honored from questing there. More reputation is gained from Emissary Quests and World Quests. You can read the full reputation guide for The Dreamweavers here.
5) The Wardens. This faction is a World Quest-only faction, and does not have any quests tied with them. They are located in Azsuna, but in order to get rep you need do to World Quests and Emissary Quests. These will most likely be the last one you get to revered unless you specifically focus on them. Read The Warden's reputation guide here.
6) The Nightfallen. This is the faction located in Suramar, and they have a lot of quests tied to them. However, these are only unlocked at level 110, so you cannot work on this reputation while leveling up. They also have World Quests and Emissary Quests for rep gain. You can also do the Withered Army Training scenario every four days for a nice reputation boost. Read the full Nightfallen reputation guide by clicking here.
You do not need revered reputation with Talon's Vengeance, so don't bother with this unless you want the mount.
The strategy to get easy rep is to complete all the available quests while always doing the Emissary Quests, which gives +1,500 rep each. There will only be one Emissary Quest available every day, but you can save up to three of them at a time, so log in and complete these at least every third day to get the full benefit of those.
If you are behind on one of the factions, just go to their zone and complete World Quests to catch up.
Some Class Hall Missions can also award rep tokens for +250 reputation with one of the factions, so keep an eye of for those.
A Glorious Campaign / Complete the Class Hall CampaignThis part of the Pathfiender achievement is unlocked by fully completing the Class Hall Campaign, and the exact quest chain is pretty different for all the different classes. In order to get this quest chain done you will have to complete a lot of regular quests in the Legion zones, as well as going into a lot of different dungeons. You start this quest chain once you unlock your Class Hall, and new quests are unlocked for each level up. Upon completing the whole chain you get an ilvl 830 chest piece, a unique title, the third artifact relic slot, and of course this achievement.
I'm sure most players have already unlocked this on their main because the third relic slot is pretty essential to playing at max level, but if not, just check your quest log and work on the chain.
Broken Isles Pathfinder, Part TwoThis second achievement is also needed to unlock flying on the Broken Isles, but is a lot easier than the first part. Below is the stuff you need to do to unlock it:
1) Have Broken Isles Pathfinder, Part One.
2) Explore Broken Shore (a new and overhauled zone in patch 7.2).
3) Breaching the Tomb (probably tied with the main quest chain i patch 7.2).
4) Defender of the Broken Isles (again, I guess this is tied with the main quest chain).
5) Legionfall Commander (tied to a certain reputation level with the new patch 7.2 faction Armies of Legionfall).
This part of the guide will be updated once patch 7.2 hits the PTR and we can find the exact stuff that needs to be done to unlock the flying mounts on the Broken Isles.