My sons grew up sharing a Savage 110 youth model in 243. Between the two of them approximately 2 deer year per year from 13-to 16. My younger son received a 270 for his birthday at age 16. From that point on he has killed elk and deer with his 270. My older son still uses the 243, and loves it. (no elk yet.)
I grew up sharing a 25.06 with my dad and brother. The gun was a Winchester model 70 purchased from Gibsons in approximately 1980-2??. To many deer, elk , coyotes, and pigs to count have been shot with that gun. My dad still uses it. From the time I could afford to purchase my own rifle, I have used a Sako 25.06. Again too many deer, pigs, and coyotes to count.
If it is primarily deer and coyote country, I would suggest 243, 25.06, or 6 CM or 6.5 CM.
If it is primarily elk and mule deer, you might consider 260, 270, 7.08, or 308.
Personally, I don't think I could have handled a 308 at age 12 or 13 very well. A 243 level of recoil is about perfect for a young shooter, IMO. The main thing you want is for them to be hooked. You would hate to see them get scared off by a heavy recoiling gun. I have seen a couple of young people who don't like hunting because their only experience was wing shooting with 12 ga. when they were too young. There are tons of good suggestions. I don't think you can go wrong with 243, 25.06, 6.5CM or many of the other suggestions.