For western pleasure, you pretty much need to wear a cowboy hat. You usually want to get two kinds - sand and black. Felt hats come in a variety of grades, "x". There are 7x, 10x, 20x, 50x, and 100x. It depends on your price range. I think I have a 10x because I'm a little poor. You also need a hatbox for your hat, as you don't want your felt to get dusty. Straw hats are frowned upon in the western pleasure arena (unless you're a man). You also need to get a hat that has the "Quarter horse crease" and you also need to find someone to shape your hat into the appropriate style of the time. Nothing looks worse than a hat with a flat brim. You need to shape the brim up squarely and sharply.
Hope this helps :)