What are the costumes?

Yes. Once you have unlocked them, go to the character select screen (Battle Only) and select the character that can change costume. Drag their icon to the slot to where they state their name. From there, go back to the character slot, press "Z" or "C" for the alternate costume and drag the character icon to the same or another slot.

They made this more difficult than it had to be.

You don't need to drag it in twice, you just hold C/Z then select character