Upon clearing the entirety of You in Wonderland, you will gain access to requests (sidequests) that you can get from Elizabeth in the Nurse's Office. If you've played either Persona 3 or 4, then you should be familiar with these types of requests, as they are mostly similar.
There are three different types, as described by the game, which are Find, Meet and Hunt. Hunt requests will require you to kill an enemy in one of the dungeons, while Find will usually have going to a specific place or doing something. Meet ones are the most involved, as they usually require you to do something related to the game's cast.
Some of the requests are actually timed, although it's not exactly clear when those requests will be taken from you, but it does appear the game will warn you before they disappear. Also, some requests will give out more rewards depending on what you do during them. For example, in one of the first requests you can get, if you do something in a particular order, then you will get a few bonus items.