You obtain costumes by unlocking them via the story or buying them from the store Croco Fur in Okina City once you get a scooter license. (To get to Okina City you first need to go on a scooter ride three times to build up enough courage to go far enough that you'll reach the city.)
Once you have a costume, go into the main menu > equip > select a character > costume then select a costume to wear. This will unlock during the next fight.