These mounts are quite easy to get. Just about 3 weeks doing Ananta Beast Tribe Quests after finishing Stormblood. These quests take about 10-15 minutes each day, except for the special quests when you Rank up. Only one of the quests is at all challenging. There is a solo duty along the way that requires some care in the battle. The first mount available is the Marid Elephant.
And then at Bloodsworn you can get the Griffon.
Of course there are other Bird mounts in the game, but personally, I like the griffon. The Lanner [Falcon] mounts are nice and usually have glowing effects, but they are “birds” and so run on two legs, which is a little jarring on the ground. The griffon runs on four legs and so has a smoother ride on the ground.
I suggest you complete Stormblood and get flight in Gyr Abania before starting these quests, then just leave your character at the Peering Stones crystal to do the quests each day. The terrain is much easier to traverse on flying mounts, so waiting until you can fly in this zone will save a lot of time.
May you walk in the light of the Crystal.