This is the quest log I'm talking about. Then I assume you have to complete [shopkeeper's sickness] to start full moon. Bernerd(shopkeeper) plays a main role in the full moon questline. Have you meet Logan(lumberjack) yet? You need to know both of them to advance in the questline. In case you haven't met him, explore the forest to find something belongs to his wife.
Finished quests includes some minor quests from various NPCs, some of them are repeatable.