Even if you don't own the new Chapter, you can start the adventure right now by playing though the Summerset Prologue quest.

You can begin by acquiring the Mage's Message Stone from the in-game Crown Store (it's free). Simply use the stone to receive a message from Vanus Galerion and acquire the quest “Through a Veil Darkly." If you're unable to acquire the stone, you can find the ancient mage himself inside the Mages Guildhalls within Vulkhel Guard, Daggerfall, and Davon's Watch (depending on your Alliance). Likely with his head down in some dusty tome.