@What are quests?
Quests are missions for you to do. They will give you a goal and instructions how to achieve it, and when you've succeeded you'll receive the reward that is listed in your quest log. Optional quests often reward unique equipment and achievements.
@How do I check my quest log?
Open your inventory (default "I" on keyboard or "back" on gamepad) and navigate to the red exclamation mark in the top right of your inventory page.
@Where do I find quests?
The main Story quest line is obtained from Korem in the Chronicon. Other quests are often found near entrances of various areas. If a character has a quest for you it'll show an orange "!" over its head. If you already have the quest it will be a grey "?", and if the quest is ready to be completed the "?" will be orange.
@How do I complete a quest?
Some quests will automatically complete when the goals are met, others will update and require you to seek out someone - this is usually the person who gave you the quest. If a quest is updated, you'll receive a text prompt on the screen, and the quest text in your quest log will be updated with additional information.