If you reload or know someone who does, I'd go with a 308 or 30-06. My son started with a 308 using 110 grain hornady R/N with 12 grains of H110. After he shot his first moose (a 300 yard shot) when he was 11 years old, with his moms 308 using Federal "high energy" loads, I asked him what he thought of the recoil and he said "what recoil". He's now 17 and has his own 338 WM, which he shoots very well. We were at the range the day before yesterday, and he fired a 30-378 weatherby. He hit the gong at 200 yards and said it was very nice.
A friend's son was afraid of his 308 and the dad asked me to load some practise loads for his son. His 13 year old son road his bike over a few days later and asked me if I could "refill" them. I loaded him up a batch and said I wanted a roast from his first moose. That fall he road over on his bike and handed me a roast.
When my son was 12, he fired a very nice 243 at the range. At that point he hadn't fired any hunting loads except while hunting. He said it was a nice rifle but kicked a lot more than his 308. 30-06 works well with 15 grains of H110.