My son started on the lawn tractor when he was 4. (never turning blades, just steering it) Back then I'd walk behind him holding the seat down to activate the safety switch. If he got going too fast or otherwise wanted to stop, all I had to do was let go; he didn't weigh enough to hold the switch down so the tractor would die. Now he's heavy enough and tall enough (6) to operate it by himself. I put him on the kubota also starting when he was 5. He can't work the clutch yet but can reach the brake and work the steering and throttle and knows how to kill it when needed. My daughter on the other hand is currently 5, and is not interested in being in or on any vehicle without a chauffer and air conditioning. She's gonna have a hard childhood.
Soundguy, my wife has driven the kubota a bit and does fine with the gears, but can't figure out/remember how to start it, stop it, or what all the other levers are for. She's not someone you want to have help you hook up a heavy implement.
My neice comes to spend the summers with us, she's 13 now. This last summer she wanted me to teach her how to drive. I started her on the lawn tractor (never actually saw someone flop a lawn tractor on it's side before) and then once she was no longer actively dangerous on that moved her to the kub. Lemme tellya, if you mash it hard enough you don't really need the clutch. Never mind the grinding.

She spent hours on it. Eventually got pretty good, but alas she had to leave before she got to drive one of the cars .
So basically, it depends on opportunity, skill, and interest. Start'em young.