Hadn't thought about the bolt gun for kids. That's a great idea Kirk. Reps to you sir if I hadn't whored them all out for cheesy posts already.CZ 452 Scout. Two important consideration for a kid's gun: weight and LOP. The Scout is light and has a 12" LOP. It also has nice big iron sights.
I prefer bolt guns for kids as you can remove the bolt from the weapon. It needs a sling for safety too.
Marlin model 60. An accurate semiauto 22 and is about $150. Put a rimfire scope on top for about $40. and you're in the game for plinking and for small game.
A bolt action .22LR is the gun for a twelve year old to learn how to shoot.
More fun than a single shot, but won't pound through ammo like a semi-auto and will be easier to develop discipline and eye for safety when manually operating the bolt.
.22 LR is the cheapest ammo by far.
Search around for the Marlin 22-XT. They are in the below $200 price category. Magazine fed, and iron sights. Can get a scope later. This has the Microgroove barrel and ProFire Adjustable trigger. Very accurate out of the box. Will hold reasonable resale value if you ever want to resell.