Myrna mentioned some things you will need for the competition, but here are a couple other things you will need on the way there:
A slow moving vehicle triangle for the back of your cart when doing roadwork. You can buy one, but you can make one a heck of a lot cheaper - just an orange triangle. I did actually get stopped by a cop for not having one! (It's a small town)
A gear bag that will attach/detach quickly. I know some people use a fanny pack and buckle it through their basket on an EE cart. Just big enough to hold water bottle/sandwich/kleenex. I share my water bottle with PJ. I also tend to come in late from work, throw together a sandwich and toss it in my bag. Sometimes share that with PJ too.
Oh, and don't put your cell phone in the gear bag. If you take it with you, put it on your personage - preferably in a heavy duty case. If it's in the gear bag and, heaven forbid, you and cart part company . . .
A spare halter and lead in case you need it while on the road.
We have tons of goatheads, so some of us carry FastFlat.
Also, look at competition photos and see what the drivers are wearing! Plan your wardrobe carefully. You will need brown gloves! This was my first year driving, I had a heck of a time trying to figure out what to wear. PJ is leggy and not very flashy (my brother calls him "Homely Boy"), so I got a brow band that would give a little colour to his face. I drive a HyperBike, so my attire was going to be a little different - sure couldn't hide that it was a "bike" not a cart. Also, I wear a helmet from eventing days, so it has a cover. I finally went with grey slacks and a long sleeved green shirt and my green and grey cover. The shirts have changed during the season, but always long-sleeved, and in greens or blues and grey, with matching cover. Our last ADT, the dressage judge scored the Presentation very high for us. When someone asked why we'd got such a high score, and they got such a low score, the judge said because my horse was rather standardbred in shape, so the 'Bike was appropriate to his build, and my attire was appropriate for the 'Bike and well coordinated. The fact that I roll my sleeves up on the marathon portion had nothing to do with my dressage score, of course. But that score was one of those little things that helped us win our division this year. So see what is appropriate attire for your area, and especially for your horse and cart. Treat yourself to a new wardrobe! (sort of).
Gather cones or coffee cans - anything you can make a course out of and work up to 20 gates. Learn to memorize a course after a few walk throughs - not just driving through randomly.
ohhhhhh, you are in for so much fun!