Read some books and watch some videos as a start. I've watched many driving horses trained and know that if you skip steps, or do things "wrong", it's way way harder later on to try and fix things than if you were able to do it right the first time.
Personally, I would get the horse used to the bit first, lunge in part of the harness, use an open bridle (NO blinders), ground drive, have the horse drag something other than the cart, have a helper pull the cart beside in front of and behind the horse while I ground drive, work up to hooking the cart to the horse and ground drive to the side or behind the cart, work up to tying the cart down on the harness and eventually getting in. It's not something that happens in just a couple of weeks and I'm not sure that having the horse pull the cart with the halter on is the best first step. (It may work great, but isn't something I've seen done -- not to say others haven't done it that way and it worked but it's different than what I know.)
I've trained one of my own horses and made mistakes not knowing better! I've learned a lot since then and probably know 10% of what someone who is really good at driving training knows. I've had many horses professionally trained to drive and am able to appreciate the difference between a job done well and one done by guesswork.
Currently, I am slowly working with one of my geldings towards driving as there is not anyone close enough that I'm willing to send him to and I know a lot more than I did when I trained my first driving horse.
I think if you are able to get some good books, watch some videos and have a person or people who know what they're talking about, you can for sure do it yourself but it's so important to take it slow, consistent and correctly. Mini are small but it would be pretty terrifying to be in the cart with one who's running off or even rearing and bucking in my opinion.