You'll unlock the ability to change your apparel during the second chapter of your adventure in Harry Potter: Hogwarts Mystery. You can wear a top, pants, and various add-ons. Unfortunately, only a single item from each group is available. Rest of the items have to be bought for money or diamonds which are amassed during the gameplay.
You can change your clothes at any moment. Some of them are unlocked only if you reach a certain experience level. If you haven't reached the required level, you can buy the item earlier by spending violet diamonds.
Colors or patterns cannot be changed. The clothes in Harry Potter: Hogwarts Mystery mainly differ through color, rather than style. If you want to wear clothes of a different color, you need to buy them.