I own a 7mm-08 in a synthetic Browning A-bolt. I think that it is just about the perfect caliber for deer sized animals. It has mild recoil yet shoots pretty flat. I haven't had a deer make it farther than 40 yards. For Elk, I would think that if you kept the shot distances within reason and used a tough constructed 140 grain premium bullet like a tripleshock ora partition then it should kill elk reasonably well. But I'm by no means an elk expert. The 308 with it's ability to handle heavier bullets would probably be better for elk but then you are looking at more recoil with that set up. You would know far better than me what level of recoil your kid can handle.

As for brands, well they ain't beautiful to look at but Savage makes some pretty cheap yet accurate rifles. You don't really need some wallhanger masterpiece with a high grade walnut stock and fancy engraving for a kids gun anyway.