Every horse can be started in "ground work" as soon as they are weaned, if not sooner. Babies can learn to tolerate (or even enjoy) brushing, getting their feet picked, etc. Halter work can start at about the same time. Obviously, you don't want to pull hard on a foal's sensitive poll, but they can get the idea about leading, stopping, and standing. Voice commands can be introduced at that time as well. It all translates to driving training eventually. What I would NOT do is turn them out to pasture and do nothing with them until they are 2-3. I have had yearlings wear the harness (not including the bridle) in the stall. You can also "play" in their mouth so they get used to the idea of something being in there.
Yearlings can also learn "showmanship" work, such as setting up, standing, trotting off in hand, whoa, without too much work on circles. When older, they move into longeing fairly easily, because it is just the same stuff only in a circle. You can introduce the long lines on the halter with the rest of the harness, too. Bits get introduced when their mouths mature, and not before wolf teeth are checked out. You don't want the first bitting experience to be annoying with bumping wolf teeth. I tend to use a mullen mouth snaffle, as there isn't much for them to "play" with then and they learn to hold it better. As I see more about what their preferences are, I might switch bits later on.
As far as actual line driving, again, that depends on the condition/maturity of the horse's mouth, but probably around 2-3. You need to keep monitoring the teeth in the process. It also depends on the maturity of their minds.
I would not put the horse to a cart until their 3rd year, and it doesn't hurt to wait longer if you intend for the horse to be sound and viable for 20+ years.