You should clip your horse the night before the show. That way, you have plenty of time, and if in the morning you find a spot you missed, it doesn't take long to fix.
You also need to clip the whiskers above and below their eyes, but avoid trimming the eyelashes. Make sure you cover the horse's eye so stray whiskers don't fly in and you don't hit their eye.
Once you're done trimming, clean out and oil the clippers so they're ready the next time you need them.
In the morning, brush all the dirt, hay, and dust off the body and legs.
Comb the mane. Tip: start from the bottom and go up (do this for the tail as well). Don't forget to comb the forelock.
If your horse's tail (or mane) is really tangled, you may want to use a detangler before you start combing through it. I use Cowboy Magic.
Comb the tail from the bottom to the top, just like you did with the mane. You can put in a fake tail if you're at a high-class show (AQHA), but most of the time you don't need to worry about one.
Clean out the hooves to make sure there aren't any rocks or pieces of metal that could injure your horse. Start at the heel and scoop out towards the toe.
Apply hoof shine to the top of the hoof and let it run down. You can fill it in later.
Make sure to cover the whole hoof!
Use a soft face brush to get the dust off the horse's face.
Apply Vaseline to the muzzle and eyes to get a shiny well-groomed look. Be careful not to use too much, or it will look greasy.
Spray the body lightly with Show Sheen and you're ready to go!
The final product will look something like this. Good luck showing!