It depends on the child. It isn't a bad idea to start them on a cheapie no contract phone. If they lose it, no big deal. And since you pay by the minute, they can't go over and trash your plan. Once they demonstrate that they are responsible enough to not lose/destroy a phone, and understand the idea of tracking usage, they are probably old enough for a contract phone.
I did this with both of my kids when they were about 13-14. My older one is good about it, my younger one just lost his iphone, again, so I got him a $10 trac phone. It can go the other way. And he's 17.