Nothing, nothing, beats grooming your horse everyday for a shiny coat. I have a chestnut as well and he gets groomed for at least a half an hour everyday, and he glows. As far a supplements go, he gets flax seed, Gold-As-Sun, Source and corn oil in his feed. He also gets a joint supplement, but that doesnt help much with your question. He is clipped when he gets furry, and gets blanketed or sheeted if it gets chilly out. As far as shampoos go, I like using Orvus. It comes concentrated and alot goes a long way. I have had the same tub now for almost 2 years and I still have a little less than a half a tub left. It works wonders too. I also just use whatever human shampoo is on sale when I am at the grocery store. For color enhancing shampoos, I have wanted to try this one brand that I have heard alot of nice things about, but I cant remeber the name. There are shampoos for every different coat color. I also use Johnson's Baby Shampoo. That helps with my chesnuts dry skin (along with some special medicated shampoo I have), and I like using it on horse's faces because it is Tear Free and wont sting if it gets in their eyes. As far as conditioners go, I pick up whatever is on sale at the grocery store. I think I have some Suave Coconut conditioner right now. There is no need to pay a bunch of money for shampoo that does the same thing as the cheap stuff. Just remeber, although bathing your horse may get them looking clean, it actually isnt very good for their coat, It takes out the natural oils that they produce and drys out their skin. Try to only bathe when you have to. If you keep your horse groomed every day, giving a bath before a show shouldnt be necessary, you just have to worry about stains that they might pick up; and with that, you can just spot clean.
I have a bay paint as well, and I did Show Grooming, Hunter in Hand, and Color classes with him, and I won every time in my show grooming class. So if you need any grooming tips, feel free to ask =]