When my brother and I were younger, our dad had a really nice, accurate .223 that he let my brother learn on. One evening, he sat with our dad and I was off doing my own thing. Dad left him in the blind and went off to do something, tell him that if he sees a deer, shoot it.
Later that evening we ended up having to field dress two does he'd shot. Following directions. We learned alot from our dad....mostly what NOT to do, heh.
I got a .308 for Christmas last year and I'm thinking about outfitting it for my 7(soon to be

year old daughter for next season. We have a little Ruger .22 I set up for her to target shoot and she took to it like a duck to water. Very careful and patient...I mean, the perfect little hunter. This is her first season so the deal was that she had to sit and spot and prove she could handle all the other stuff with hunting (being quiet, not getting bored, etc) and we'd work on a good rifle in the off season. so far so good.
I'm thinking the .308 would be good for her. My younger sister came up with one and my brother still has his he started using as a teenager. It my kick a little, we'll try it and if that's a little much (my daughter is very petite) we'll go with a .243 or see if my dad still has the .223.