Quests in the game can be divided into 2 types: main quests and event quests.

Main quests: completing these quests is required to progress through the game. Completing each quest gets you 1 collection item. Therefore, your goal is to solve 5 puzzles or explore 5 locations, then combine the items you get into one collection.
Your active quests can be found on the left side of the screen. New quests are marked with a green exclamation point and a green arrow pointing at the quest.
To get more details about a quest, press its icon once. All quests come from Darkwood residents.

Event quests: a new event starts with every update. Each event has its own location, items, and fixers.
Event quests are not connected to the game's main storyline, but they have their own plots and give players a great chance to win special prizes such as unique avatars, jewelry boxes, desk guardians, talismans, energy, tools, and chests. In order to obtain prizes, you will need to fulfill the required conditions of the event.
Even after an event is over, all of the collection items and fixers related to it will be available at the Store.