Currently there is not much to do in the free mode, i.e. you can talk to the characters and look for the freeroam events around town (The fairy in the MC's room tells you where they are), you can collect the shiny things to unlock the memory events, but otherwise there is nothing else.
Inatari left a little aside that section while advancing in the main story of the game, but soon we will have more content for the free mode, we just have to wait for it.