Quests are adventures and tasks you can undertake to get treasure and experience for your Champions. In order to start a quest you need to move a unit onto a quest location. Once you start a quest the next objective you need to complete will be listed in your quest tracker on the left side of the main UI. If you ever get lost you can click the icon to focus on the next step in the world. Each quest will require you to do different things to complete them so get out there and see them all.