For local Open shows, you can start out with nice jeans, a nice western shirt, belt, hat, and boots. If you are going to a bigger Open show or a breed show, you should invest in a rail shirt and chaps. For breed shows, rail shirts and chaps are a must. If you have a chance, go to some shows as a spectator before you actually show. That will give you an idea of what to wear and what the show will be like.
I will say that I am a big fan of the lycra rail shirts that zip up the front. They will give as you ride and move, they stay tucked in better than cotton western shirts, they don't wrinkle, and you don't have to worry about losing a button or having the shirt come unsnapped as you ride. I think they give the rider a more clean, polished look. (JMO) You can find some good starter rail shirts on eBay
for around $50.
Good luck and have fun.