Go to www.NATRC.org
for details of North American Trail Riding Conference. They host Competitive Trail rides across America. If they don't hold events in your area, search for other organizations that hold similar events in your area.
Competetive Trail Riding is more like a Road Rally. It's an event, ( here in the west they are usually weekend events.) Where you and your horse compete over a couse. Usually 40 miles for Novice and 50 miles of Open classes over a two day ride. It depends a lot of where the event is held, how rough the terrain is etc. The novice course will be timed at a slower pass, while the Open class will move at a slightly faster speed.
Horses are judged on their conditioning and ability to handle various obsticles. Riders are judged on their horsemanship skills. Judges will observe you before the ride, during the ride and at the end of the ride. Usually a vet will check the horse for metabolics such as Respiration, heart rate, hydration and gut sounds. They will also look at the horse at the end of each days ride looking for lameness, swelling, filling etc.
You as a rider will be judged on how you handle and care for your horse.
If your horse can handle a good 15-20 mile ride with out problems, He should be able to handle a Competitive Trail Ride. Being able to do a brisk walk with occassional stretches at a trot will be all you will ask him to do.
Competitive Trail Riding is NOT a race. All riders will complete the course in the same time. Too fast or too slow and you are out of the game.
Find a event nearby and go enjoy. If you are worried about it. Go and volunteer to help for one ride to see what goes on. You will have a great time.
Prices will vary a LOT depending on who is hosting the ride. Land use permits, Forest Service fees etc. But you should expect to pay $70-$100 for a competitive trail ride.
Everybody at the ride will be willing to help. Lots of rides in the West offer Mentors to help novice riders on their first few rides.