At the start of the game you can choose one from 3 different styles:
Later in the game you will be able to change the outfit (hat, shoes, and accessoriese) and the hair style of your character.
To do so you have to use the boutiques and the salon that appear in the game. There are boutiques in many of the cities you visit and each one has its own individual style.
When you change your outfit and your hair, your icon on the PSS and on the Pokémon Global Link will change, too.