The Album menu is where you can find all the pictures of the girls you've unlocked or have yet to unlock.  You can sort them from the right side for each specific girl, and you can even favorite photos you like!

The Quest menu is where you can find all the existing types of quests in the game.

There are three types of quests : the Daily ones, which reset and can be done multiple time, the Main and Achievement quests which are one time only, and the event quests. Visit often to check if you've unlocked any!