In Balan Wonderworld, changing in and out of costumes will be paramount to finding all collectibles and progressing through each level. The costumes essentially play the role of power-ups in the game, with each individual costume giving your character a single specific special ability. With that in mind, here is how you can access the dressing room and swap out your costumes at any time.
To access the changing room, all you need to do is find a checkpoint in a level, stand on it, and press any button, since all are action buttons. If you have not cleared the level before, only one option will appear, which is the changing room. Otherwise, four rooms will pop up, the changing room being the one on the far left. Simply walk into the changing room to begin.
When you are in the changing room, you will have access to all costumes you currently have available to you. If you look at the number next to each entry, that will tell you how many you have of that kind. Unfortunately, the only way to refill your costumes is to go to a level where you can obtain that costume and grab as many as possible and go to a checkpoint to store them.
In the top left of the screen, you will have three slots where you can choose any costumes you have available to you. Select any order you want to cycle through in the level, and you can kit yourself out with them to take advantage of their specialties.