I got into the Alpha last night. No idea why, but I've been questing in Tirisgarde Sound for the last six or so hours. Only at level 112, but I've done most of the zone so far. I still have at least one more major questing hub to go, and the conclusion of the main storyline. I did find a quest that just sort of ended without letting me complete it. I has to close the quest tab via the X with no followup. It was part of the Alliance War Effort, so I don't know if that is fully in game yet, or if it picks up later. That is where the mission table is, and that's suppose to tie into Warfronts and I suppose going to the opposing faction's territory.
Didn't do any introductory quests, it just put me in Boralas with a chest piece of Azerite Armor that has minimum customization options (first tier is basically spec based or pvp.) My Heart of Azeroth is already at level 5, so I imagine it will hit ten long before I hit max level (currently capped at 119, I imagine they'll add 120 once they release the Warfronts and want to test the World Quests)