In order to begin any of the sidequests related to the characters, you will need to have gotten to the end of the main story, pretty much. After defeating the Time Judge, there will be two things that become available. The first is a bridge that leads to the final boss, and the other thing is a portal on the right hand side. This will lead you back to the entrance of the Last Lands, with the portal there leading you back to right before the final boss, which will allow you to leave to the outside world again.
It is at this point that you can undertake the sidequests. Every single character will have a sidequest available and will require you to speak with someone in the world. The only catch is that the character whose sidequest you’re looking to do will need to be in the active party in order to start the sidequest. Each sidequest will reward you with the strongest Spritnite for that character, which will also result in a system trophy.