Driving can really get your horse in shape, and used to the outdoor world (away from a barn and horsey friends).
I sent my mare to a trainer, and he trained her to drive in 2 months. She already had good ground manners though, and had been desensitized to quite a bit, so half of his work was already done. She's also a very quiet mare-for the most part!-and acted like she'd done it every day of her life.
Unless you're a leather worker with years upon years of experience, I wouldn't recommend making your own harness. Same with a cart. Don't get me wrong, it can be done. But personally, I want to drive something that I KNOW won't fall apart!
There is actually more to driving than just hitching up the horse and making it pull you around. Many people drive horses like they are doing dressage. My trainer did not train my mare in this way- he just taught her to pull the cart around without spooking at everything. Training your horse to move correctly while pulling a cart is more difficult, and takes more time. Driving IS a very neat sport, but one that you need to be completely competent in before embarking out on your own. I only drive in an arena, because I KNOW I wouldn't be able to handle a situation if one arose outside of the arena. There's nothing wrong with this, and maybe someday I'll leave the arena. But as of right now, there is no safe place for me to drive, so I stay in the arena.
It's funny to see how different people approach driving. I let my boyfriend drive my mare in the arena one time. He took the reins, slapped her back with them, and yelled, "HEEYAH!" Thank god my mare was understanding, and didn't take off. She just gave him this look like, "Ugh! Amateur!" Lol! I had to tell my boyfriend, "Never do that again. Just take the reins up, and tell her to walk on." He looked quite sheepish after, and said, "Oh. Sorry. That's how they do it in the movies..."
Good training is essential for both the horse AND the rider.