What do you have to wear when going western horseback riding?

This entirely depends on if you are showing or just out for a leisurely trail ride.

For a trail ride you'll want jeans with no inside seam on the legs, boots with a low heel, a long sleeve shirt (thin in summer, thick in winter.) Socks, and a helmet. Although it is customary for western riders to use a cowboy hat, it is not safe to ride in anything other than an approved riding helmet.

The clothing for a show will vary somewhat depending on which class you are competing in and it is best to consult the rule books for that show or division.