There are a lot of good rifles that will do what you want. Make sure the rifle stock fits your kid. Probably the best buy on an entry level rifle right now is the Marlin X7. The Freedom Group,(Remington) is stopping production on the rifle and it can be had in a lot of places at a great price. Davidsons, (Gallery of guns) had them last week For $250.delivered to a local FFL. They are chambered in 270 Winchester and come with a one piece Weaver style mount. They also have a reasonable recoil pad. The couple that I have shot have been very accurate. I don't know how much you are into working on rifles but they are a barrel nut rifle and can be easily be rebarreled at home. In this case, it could be made anything that shared the same base as a 270 simply by screwing on the new barrel and using headspace gauges.
