To begin Dragonborn, you must travel to Solstheim. This can be done at any points after installing the game add-on content by travelling to the Windhelm docks and locating Gjalund Salt-Sage, a sea captain that may be willing to take you to the island for a small fee.
The “main” questline of Dragonborn will start soon after you have started the main game’s quest “The Horn of Jurgen Windcaller”, after consulting with the Greybeards for the first time. After having started this quest, you will soon run into a group of Cultists when traveling around the world that will discuss Dragonborn’s questline.
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Make sure that Skyrim is updated with title update 1.8 or higher. If you continue to experience issues, you might want to re-verify the Steam game cache files