You can accept quests one at a time. However, it can be increased up to three by acquiring the “Quest Accept Up” skill. These are sidequests entirely different from the main storyline so you can take them to earn extra exp, Blood, libra, and items. Always check what quests are available with Simon whenever you're back in the hotel especially after using a Door in the middle of a scenario. Some quests are repeatable after some time has passed. There are three main quest types: Night, Day, and Event quests.
Night Quests
These are the common quests that you'll tackle since you'll be doing them while you're fiend hunting at night. You can accept multiple night quests after acquiring the proper skills for it. Night Quests are divided to three lesser categories: Slay, Search, and Investigation. Slay quests will require killing a specific number of a specified type of enemy. Search quests will give you a task to find a specific item in a designated area, and Investigation will require you to visit the indicated location.
These are quests that can be set as a Daytime Activity. They'll be completed automatically along with other Daytime Activities without further actions from you. You can only accept one at a time.
Event Quests
Some of these are unmarked quests and will appear only as events. There are no visible reward or completion prompts but the game will indicate if you're tackling it already or not. You can also check the available tabs in the Scenario menu.