There are six costumes that the boy can change into, one of them by default. Most of them are cosmetic, though one does provide a use. In Chapter 2, at the very beginning, there is a trophy that requires you to pass by the shades undetected. However, it is misleading, as it is simply talking about the shades you can't interact with when you first awaken in the chapter. To satisfy the requirements, you have to equip the Shade costume which is first found in Chapter 4 (so you'll have to use Stage Select to get back here afterwards).
Somewhat Awesome Games has provided a video of where you find the costumes in each chapter. I will go through and timestamp the specific length of time you should pay attention for the chapter you're curious about.
*Costume #1 is default, yet for numerical sake since you can get them out of order, I am going to classify them as how they would appear.
Chapter 1:Costume #2: 0:06 - 1:35
Costume #3: 1:45 - 2:35
Chapter 2:Costume #4: 2:42 - 3:59
Chapter 3:Costume #5: 4:05 - 4:48
Chapter 4:Costume #6: 4:57 - 6:18