To unlock costumes you'll need to collect crystals. Once you have enough crystals you can unlock costumes just like this:

1. Open the crystal chest (bottom right corner) to open the costumes menu.

(Optional) If the crafting menu was open previously, then the crafting menu will open. Tap on the costume button on the right.

2. Navigate through the different costumes by swiping or tapping the arrows. It will inform you how many costumes you have unlocked and the green icons below will also indicate the information visually

When you stumble upon a costume that catches your eye, and you possess an ample amount of crystals, simply tap the unlock button to claim it as your own.

Notice how before purchasing the costume, the image will appear as a silhouette.

3. Once you've unlocked the costume simply suit up!

...and that's it!! You've unlocked your first costume.