DO NOT let your horse run away with you. Make sure he/she knows that you are in charge, and you will decide how fast he's gonna go and which way he's gonna run.
A previously mentioned, stay away from the cloverleaf pattern! Get a few barrels and make all sorts of wacky patterns to practice on, keep your horse guessing! I generally do a figure eight pattern but I like to do the Speed Barrel and Sidewinder pattern as well.
Also previously mentiioned, rate the horse. Make sur ehe doesn't take off with you every time he sees a barrel xD
And for the love of god, sit down on your turns. I see so many kids up and ut of the saddle while their turning now that its ridiculous. Sit, lean, and turn. Keep your horse supple to your seat, leg, and vocal cues so you don't have to be hanging on his mouth the entire run. That's how barrel horses get hard mouths.
Ay other questions? Ask em' here ^^