First off, if they have had absolutely NO training whatsoever, you need to desensitize them to anything and everything! Because if you don't now, someday you will be driving and your little mini might spook, and when he does spook he is going to take off running (speaking from experience).
Then, like others have said, ground drive until he is able to stop, stand, back up, right/left, start out at a walk, pick up the trot.
Once he is broken from ground driving, you can start showing him the cart. When they first meet, the horse is going to be more afraid of the cart. So let him have time to check it out. Then start walking your mini right next to the cart (you have to pull, sorry) just in a simple halter n lead, and do all of your basic maneuvers.
After that you need to make sure he is used to the bumps/the way the cart moves. So pretend like you are going to hook him up, and put him in-between the shafts. Then just kind of bump the cart around, not really hard or anything, but when you drive the cart is going to hit him a few times, and you only do this so that he builds up more trust with the cart. (i know, carts are just absolutely terrifying!
After he seems to "be friends and trust" the cart, then you can actually harness him up to the cart. Now, when you get to this part, still ground drive, but bring your reins up and over the back of the cart.
Finally, after he is completely desenitized to the cart, and actually listens to your commands/cues, you should be all set to get in the cart. Though I would suggest for your first time taking him into an enclosed arena just for safety.
And there you go(: Sorry I practically wrote a book, I have just learned the hard way when it comes to minis + driving.