Wow, this is the place to come for questions like that.
There was an entire thread some time ago called "Pet peeves" or something like that by Chamomile; look it up! It turned into this fantastic thread with pictures on the fine points of harnessing correctly. You'll learn lots. There have also been some great "how to" sites linked before for newbies. Let me see if I can find the one I remember....
Ah yes, here it is.
Regency Miniatures "How To" page
That's the introduction page, if you go to the "Mini Driving 101" index on the left you'll find wonderful descriptions of how a harness is put together and how to put it on your mini. Highly recommend.
Liltnt, yes, you can start with pieces. If you horse is not broken to the bridle and having a bit in his mouth it is important to get him used to that before anything else. If he doesn't know how to stop and turn from your rein signals there's no point going further!
The chest piece is called the breastcollar, and you won't need that until you start him dragging weight. For absolutely basic training and beginning ground driving, all you need is the bridle, reins, and the backsaddle (the part that goes behind their withers and has a cinch.) This is so you can run the reins through the terrets which keeps them off the ground, on the correct side, and coming from the right direction if the horse spins around on you (which he likely will at first.)
There are better people to ask about how to start a driving horse, but I'd say first get him used to a bit. Teach him to open his mouth when you put a finger in it, reward him. Repeat until he opens calmly and immediately for pressure on his bar. Then get him used to having something slide over his head. Practice with a loose halter or something and be careful not to squish his eyes. Handle those ears respectfully too. I see so many minis who just freak when their bridle is put on because the handler just shoves it over their ears with no care at all because the mini can't raise its head high enough to get away like a bigger horse can. How would you like that??
Then put something tasty on a good bit and slip the whole bridle over their head and into place. The horse will probably gum it to death- expect that and let them. DON'T leave the horse unsupervised or tied up by the bit! Never never never. On a big horse I would leave the bridle on them and them loose in a small pasture for several hours, but I wouldn't do that with a bridle that has blinkers. Too easy to rub off or get hurt.
Repeat this until the horse accepts it calmly and then attach the reins. Have someone snap a leadline to the bit, or better yet put a halter on over the bridle, and have that person lead the horse while you signal them with the reins. Be very gentle and reward any effort by the horse to give to the bit with an immediate release of pressure. You want them to find out it's a good thing to turn in the direction of the pull. From there it's fairly obvious. Drive them in circles, serpentines, stop, walk, turn again. Have your handler do less and less until they are simply walking beside your horse on a completely loose lead and then have them start dropping back until the horse can't see them. Teach the horse to go forward anyway. At this point you may want to remove the handler entirely and just ground drive the horse yourself. Work on standing, and especially on whoa. Get them used to being petted by someone they can't see. All during this time or before you even start ground-driving you can be sacking them out with the harness and getting them used to having straps tightened around them.
Apollo, I know this isn't what you asked but if you don't know how to put the harness on you'd better assume your horse doesn't either and start from the beginning. The more YOU understand, the easier he will learn. And I promise, it's really not as confusing as it seems!
The secret to correctly fitting harness is to always consider what each piece was meant to do. The breastcollar is for pulling- you want it to be snug when the horse is moving forward and not draping down and hanging in big dangerous loops when the horse is standing still. It needs to be above the point of the shoulder so the horse can move, and below the airway so the horse can breathe when his head is lowered pulling. The saddle is not meant to hold anything down or on like a riding saddle so the cinch only needs to be tight enough to not slip and slide. The reins run through the terrets (the big loops on the top), the shafts of the cart sit in the tugs (the leather loops hanging down on each side.) Those long straps on the bottom of the cinch are called the "wrap straps," and you snug them down around the shafts like that website shows you. The tugs keep the shafts sitting level, and the wrap straps keep them from flipping upwards when you put your weight in the cart.
The cart itself should be balanced so there is no weight on the shafts when you sit in it. Have a friend hold the shafts horizontal to the ground while you get in so you can test this. The shafts should ideally just float in the tugs while the horse pulls from the breastcollar. The breeching (if you have it) is meant to hold the cart back while going down hills. You don't want it so tight it's rubbing the horse's rump when he's moving, but you don't want it so loose that the cart runs over the horse before the "brakes" come into play. About a hand's width between the rump and the breeching is the standard I believe, measured when the horse is standing on a level surface. The breeching should be below the point of the rump so it doesn't ride up under the tail, and high enough that it rests on the fleshy portion of the horse's butt. You don't want it in the hollow above their hind legs, it can scoop their legs right out from under them!
You can experiment with your own body to feel what this might be like. Have a friend put a lead line behind your thighs at different heights and then try to resist the pull. You can do it a lot better when it's up on the muscle and not right behind your knee, can't you?
The crupper (the piece that goes under the tail) keeps the saddle from slipping forward under pressure from the cart during downward transitions or hills and also holds the breeching in place. You want this tight enough to do its job when the horse clamps his tail, but never so tight that it pinches the horse. I find there's about a finger's width between the bottom of his tail and the top of my crupper when mine's adjusted properly. I have small fingers though.
Ya'll can worry about how to properly hook to the cart later. Right now I hope that's enough for you to ponder and maybe figure out which piece of leather goes where and get your horses used to wearing it!
Do check their teeth, but also put the harness on them over and over and have a leatherworker make any obvious adjustments that may be needed before you try to hitch. That's why I mentioned so much about fit. Don't hesitate to post pictures here! I know a lot of the CDE folk are off competing right now, but there are still enough of us around to critique if you need it. Look up old threads on harness fitting too, there's some great stuff. That's how I learned- I've only been driving a year.
(I got perfect tens on my first presentation score at a combined driving event though, and I was one of the few the judge let pass without some comment on how my harness needed to be fixed or adjusted.
All she eventually said after much examination was that my horse was cute and my turnout lovely! LOL.)
Best of luck!
Leia