The War Within Beta officially started on June 5 with a lot of game-breaking bugs that won’t let players advance. A hotfix was implemented on June 6 without a patch.
One thing testers noticed is that the Tauren NPC in Stormwind and Orgrimmar was removed. This NPC provided options to teleport to any of the new zones in Dornogal.
Removing the Tauren NPC means there is only one way to get into The War Within questline. You must do it by completing the pre-launch questline which starts with the quest: The War Within. Ignore all other quests that pop onscreen.
You get this “The War Within” quest when you arrive to Stormwind or to Orgrimmar. Once you accept the quest, head over to Silithus using the Caverns of Time portal in the Portal Room.
Follow the entire questline until you arrive at the Guardian room in Dalaran. There you will see Khadgar, Anduin, Jaina, Thrall, Moira, her son, and Magni on a bed.
Complete Moira’s quests. Eventually, you will see an NPC nearby that gives you the option to continue to the expansion’s questline in Isle of Dorn. Below you can watch a video showing all the steps to complete until you are given the option to start the questline that takes you to the new continent: Khaz Algar.
Note: Keep in mind that until the bugs get ironed out, each time you teleport or have a loading screen, you will get disconnected from the game. Sometimes you will get rolled back a couple of minutes. Sometimes you will get “Character Not Found” or “A Character with that name already exists.” Keep trying or wait a few minutes.