You can't change costumes in story mode but in arcade (and probably every other mode) you first pick a character with X then up and down swiches costumes!!! You start with 2 for each character and get new costumes from completeing arcade modes. I got 6 costumes for my favorite character Ayane!!!
It can be either Arcade, Survival or Time attack. You get one costume after completing each mode until the characters' maximum. Make it easy on yourself & simply start on Rookie in each & work your way up on the difficulties until you reach the characters maximums. If what I read on the other systems forums is true here are the characters & the maximum costumes. Not counting DLC.