A .308 in a light rifle can kick surprisingly hard. I remember a little Ruger International that I gave my twelve year old son that made us both want to cry. We unloaded it, and he did brilliantly the next several years with a Steyr .243. In spite of the opinions of some of our down under friends, the .243 with a 100 gr premium bullet is an outstanding deer rifle.
Indeed, a 100gr .243 lead core bullet, or an 80gr TTSX from a .243 is the hammer of god. My kids all started with .243 and we are hunting the very largest whitetail deer in the nation in this area. Smoked them all. My daughter killed a black bear with a .243 at age 8 as well. It didn't go 3' after the shot.
The .243 is an incredibly powerful gun that somehow has been brushed with a negative bias I will never understand. The pre-64 Winchester Featherweight was commonly made in this caliber. For about $1000 you can get a appreciating or at least stable price gun rather than a depreciating asset that rots away.
After that, my kids upgraded to a 7x64 Brenneke which is a heck of a lot of firepower. Its 95% of the capability of a 7mm Remington Magnum but it has the recoil of a .270 or a light .308. It's a great compromise between lethality and recoil. My math says it was producing 14-17lbs of recoil. Mannlicher Schoenauers in this caliber usually run $1200-$1400, often including claw mounts and good German glass that would cost $3000 or more to have installed in today's money.
Another terrific couple of calibers to mention:
Vintage high-end guns in .257 Roberts sell pretty cheap, all built easily on the 7x57 mauser. That is a wonderful cartridge a child will never outgrow. The recoil is nothing. Lots of sporters on Spanish, Mexican, Chilean, Argentinian, and FN mausers in this caliber.
7x57 Mauser is another terrific caliber that produces about 12-15lbs of felt recoil.
250-3000 Savage. I believe about 8lbs of felt recoil on average.
6.5x55 Swede. About 9-12lbs of felt recoil .
These are the optimal guns for kids, women, and the recoil averse. You also have to really work at it to buy one of these calibers in an ugly, low-quality gun. Usually they are 2nd hand guns of awesome quality for roughly the same price as modern plastic junk from Cabelas.