I ALWAYS ALWAYS advice people to have a professional train a horse to drive unless you are HIGHLY experienced in driving. I've been driving for many years, I compete regularly in combined driving, I'm always 1st or 2nd in year end placings in combined driving, and I still will NOT train a horse to drive. Ride? No problems. Drive? NOPE! Driving is MUCH more dangerous than riding. I do know HOW to train one to drive, and I will often teach them to ground drive, but as to the cart... nope, I leave that up to my trainer! :) JMHO of course.
Carts can be cheap or super expensive. All depends on what you want to do. A simple 2 wheel easy entry runs about $500 new. A marathon cart for CDEs runs in the thousands (try $7000 or up! Can find some cheaper, but...). Harnesses, again, it depends. I would say roughly $500 to $1000. Zimmermans is good quality Amish harness, leather or biothane (I HATE betathane and do NOT use nylon ever!), their number is 717-354-5667. I don't know what their cost is for horse size, I've only bought mini from them (I happened to win a horse sized synthetic Smuckers Delux in a raffle, and my other two cob/horse sized harnesses I bought used).
I would ground drive in the bit I would be driving in. As to wether you can use your riding bit... well maybe some but certainly not many. A lot of driving bits are half cheek to keep things out of the way and make sure the bit won't go through the mouth... just really depends I guess. I've never used a riding bit to drive in. I have used a driving bit to ride in though hehe ;)
Never driven a gaited horse, sorry! :)