Time to Fly

You need to do the BfA intro questline and get the achievement Welcome to Zandalar to make Beast Handler Kalatak finally show up. (I was on a new toon, so I spoke to Nathanos in Zandalar and had to go back to Orgrimmar to get the questline going.)

This quest popped after taking the Orgrimmar portal to Zuldazar. It dumped me on the pier instead of the portal room. Now every time I use the Zuldazar portal in the Orgrimmar portal room, I get dumped back on the pier. Also, the NPCs aren't all in the Great Seal anymore - instead it's all set back to just before Talanji's coronation. Other Horde toons that did not do BfA also get dumped on the pier, but the NPCs (Talanji, the innkeeper) are still correctly located in the Great Seal. So...yay mount, but can we please put everything and everyone back where they belong? There was no need to screw it all up by changing where the portal takes you - which should be, you know, the PORTAL room.

Starting with Patch 10.1.5, the Riding Skill will be learned automatically by characters at certain level thresholds for free.
In addition, by reaching level 30 and visiting Boralus (Alliance) or Dazar'alor (Horde), a quest will be automatically received to learn a new flying mount, either [item=206567] (Alliance) or [item=206566] (Horde).