The Beast tribe quests are easy, they just are time gated.
There are tons on the market board.
The firmament, which you should have access too soon I believe, has some
There are mounts you can buy with achievement points.
You can buy some with MGP at the gold saucer.
Deep Dungeons has a couple

There are really mounts for nearly any sort of content.
You can find more here https://ffxiv.consolegameswiki.com/wiki/Mounts