Saddlebreds that are in training barn all are trained to the cart, as they are worked to one each week, even if they are not Fine Harness Horse, or Pleasure Driving.
It is always good to have a horse that will drive too.
Be careful in your training, and take no shortcuts with your harness and cart/wagon, but buy the best you can, and maintain it well. Much more can go wrong if harness or wagon/cart fall apart than if you are under saddle.
Make sure that you are taught by someone who is well versed in driving, and knows how to properly fit too.
Shafts for instance, that are too long, will become irritating pretty quickly to the horse, when it is trying to turn, and can cause resistance, or pain issues.
Horses make me a better person.