Auggie frequently wore an astronaut helmet in the past, and also wears a "Scream" mask on Halloween. Masks symbolize Auggie's desire to hide from others. He feels freakish and thinks people don't like him because of his appearance.
Auggie mentions shoes a number of times. This shows that he spends a lot of time looking down, not making eye contact. This is probably because he is self-conscious and doesn't want to see the reactions people will have to his face.
Auggie cutting his braid off symbolizes personal growth and a clear transition from one part of his life (homeschooling) to another ("real school").