Becky - If you go back to some of the links I posted...there is some good info there on different disciplines. Here are some others:
Eventing (interesting....I went from loving this sport to loving to compete in human triathlons as an adult
),
Dressage,
Hunter Jumper (this was my main focus all my life, and competed in college),
Barrel Racing,
Reigning Horse....these should get you started. I would encourage you to do some web searches, and look back at the links I posted before, like this...it might help with your searches:
http://www.equine-world.co.uk/horse_sports/index.htm,
http://www.horsekeeping.com/horse-training-care-info.htm,
http://equisearch.com/horses_riding_training/sports/ . Basically, all disciplines will have some type of competition. It really will depend on the style of riding you choose, which will ultimately be something you decide for yourself.
Also, your first few lessons, your instructor will probably have a horse in mind for you, as opposed to you picking one out. There are some really bullet proof lesson horses, that help you learn, and allow you to make mistakes on them, without getting tossed off. Let your instructor guide you, and teach you all the basics. How to approach a horse; which side you stand, lead, mount on; the parts of the bridle, saddle, horse's body; how to groom, etc. When I was in high school, I taught some summer camp kids, and the first thing I did was go over the basics from the ground for the first part of the lesson. Anyway, when is your first lesson??