I always show in the riding classes with a helmet (normal style, not those giant Western hat helmets). For the in hand classes, I choose a cowboy hat that compliments my outfit. A lighter colored hat is going to allow the judge to see your (hopefully smiling) face better. Dark hats tend to cast shadows, but do seem to match more outfits.
I'm not sure about the big shows, but in a smaller show, a judge once told me "if a judge ever tells you that you can't wear a helmet, that show isn't worth going to."
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.
nothing to do with showing that's just basic fashion. Can't wear straw after labor day. Everyone knows that.
That's old school my friend. Straw hats are fine to wear any time of the year. However a quality felt is nicer and more upper level. Usually you try to stick to sand (beige) or black felt hats, matching a hat to your outfit is not really done anymore either. If I am wearing a light outfit, I will go for the light felt, if it's darker, I go with the black felt, but you can mix it up as long as it compliments the outfit. A very nice outfit I saw a fellow competitor wearing was all black but with some white trim on her shirt, she wore a light felt hat with it, looked great! Would have looked good with a black felt as well.