Changing costumes in Peggle 2 seems complicated to begin with but once you know how to access the screen, it is very simple.
First, proceed to a level as you normally would by choosing a set and a level to try. Here, the character selection screen will appear. When you are in the process of selecting your character, you can press Y/Triangle to change their costume. Pressing the button will scroll through the available costumes one at a time. Once you have chosen your desired costume and character you can proceed to the level itself.
That is how to change costumes for characters in Peggle 2.