My father moved me from a 22LR to a 30-30. It is a good deer round with little recoil.
I think a 223 or a pistol caliber lever action would be good bets as well. I would not start a kid off on a 30-06, while the recoil is not super bad it may be more than some body that is use to a 22 could handle.
"I ask, Sir, What is the militia?" "Its the whole of the people. To disarm the people is the best and most effectual way to enslave them" -George Mason Co-Author of the second amendment.