For what you want to do, get started, there are some things to look at in buying a harness, and a 2-wheel cart.
The harness saddle should have a tree in it, which prevents any pressure on horse spine, like a good saddle. Saddle of the harness should be wider, 4", for use with a 2-wheel cart. Straps should be wider, not refined as seen in the QH show ring. You need a harness with breeching, the wide strap going around her rump, which acts as a braking system for halting the vehicle. Reins should be longer than length of the horse, so you can sit comfortably in the vehicle with extra length behind your hands to not lose hold easily. You will want a wide breastcollar strap, probably with a V shape for QH with low set windpipe. Easier to pull into a wide strap than a narrow one that digs in.
Bridle doesn’t need a check rein to hold her head up, which is a common feature on less expensive harness. Most check reins do not allow a horse to relax, hold head with neck level or even below her chest, should horse have a heavy load to pull. Check rein will NOT prevent her from kicking, so it is not a safety feature either, which a lot of people think it is.
An additional piece of equipment recommended for beginner horses, is a kicking strap that goes across the hips or croup, down to the shafts and DOES aid in preventing kicking. This strap is sold as an option, you have to buy it separtely from a harness seller or harness maker. With the kicking strap in place, horse has to pick up the WHOLE vehicle weight, which is very discouraging to the horse. Often nips any kind of kicking before it can happen, so you may not even notice an issue with your horse. Benefits of no hooves flying by your head are GREAT. Here are a couple photos of a kicking strap in place on a young horse.
http://s1355.beta.photobucket.com/user/goodhors1/media/Kicking%20Strap/IMG_0537.jpg.html?sort=3&o=1#/user/goodhors1/media/Kicking%20Strap/kickingstrap.jpg.html?sort=3&o=0&_suid=135933839251908997420561014204
http://s1355.beta.photobucket.com/user/goodhors1/media/Kicking%20Strap/kickingstrap.jpg.html?sort=3&o=0#/user/goodhors1/media/Kicking%20Strap/IMG_0537.jpg.html?sort=3&o=1&_suid=1359339162211004312665071639993
You will want some kind of 2-wheel vehicle to start your animal. They are MUCH less likely to get a new animal in trouble because they go forward and backward right behind the animal. A 4-wheel vehicle has the hinge with front axle, and can jack-knife if backed fast, crooked, so it can turn over easily. 4-wheels are for more experienced animals in Driving, after a good training time in 2-wheels. As you develop driving skills, you have a lot less to deal with using 2-wheels than using a 4-wheeler.
Your 2-wheel vehicle should have little weight on the shafts where they go thru the loops on horse saddle. This weight can change GREATLY with movement of the people in the vehicle, going uphill or downhill. But for riding along on flat surfaces, you want the shaft weight light on your horse, not as tiring or hard for them to manage. Shafts should ride level in the loops on saddle, not up or down if the vehicle is the correct size. As a Driver, I prefer the seat high enough to see over my horse’s back when we drive. This lets me see ahead for traffic or things in the road to deal with. Having to lean sideways to see around horse gets VERY tiring during a drive.
The most inexpensive carts (carts are 2-wheelers ALWAYS) have usually got lightweight bicycle tires and spokes. These kind of wheels do not take much to lose their air, bend sideways, because they are not designed for sideways force that Driving gives them. They are most suitable to groomed conditions of an arena, level ground. I do not suggest buying a vehicle that has these kinds of tires, they don’t hold up to stress.
I would suggest you get hold of some local Driving Club members through the ADS site above, find out when meetings or activities are going to happen. They can also be helpful in finding suitable equipment that is used, less expensive than new. Maybe help you learn how to size your animal, so you buy correctly fitting harness and a cart with examples of animals you can look at.