Both 243 and 7mm-08 work.
My sons have been hunting with me for years. #2 starts this year on his own as an adult with 6 yute seasons under his belt. #3 son is 10, and is finally deciding he's ready.
I made some simple rules above and beyond what KY law requires:
1) My sons had to have passed Hunter Ed before they could hunt. #2 did it when he was 10. #3 passed his at 8. That gave them something to work for, and it made sure they understood what they were doing.
2) No Yute rifles. My kids are big for their ages. #2 went three seasons with a regular Marlin 336 in 30-30 and then switched to a M1 Garand. #3 is starting with the Marlin 336 this year.
4) In order to hunt, you have to be able to hit a pie plate 3 times offhand at 50 yards. With #3, that was his big stumbling block, but he's conquered it now. #2 on the other hand shot as well as I could with the Marlin 336 from the get-go.
Here are some things we tried along the way.
SKS-- 7.62X39 with a 125 grain soft point is plenty for deer at short ranges. It's a semi-auto, so the recoil is diminished. The stocks are short.
Mosin Nagant M44 -- I reload in 7.62X54R, so cooking up a 30-30 ish load was no problem.
20 GA Rem 1100 -- Semi-auto tames recoil.
Mini-14 in .223 Rem: Both the SKS and the Mini-14 ended being good transition weapons for the kids. Neither rifle has actually been out on a deer hunt, but the kids used them a lot, building up good safety habits and good marksmanship.
Some people don't like the idea of a giving a semi-auto deer rifle to a kid. However, In KY the adult has to be able to take control of the rifle at all times anyway. I looked at how much time both kids had spent with the Ruger 10/22 and decided semi- operation was not a big deal. All of us only hunt with only three rounds anyway. Working with a semi in a controlled environment got both my sons very aware of the state of their rifle. Levers and bolts ended up being simple.
#2 son could not wait to hunt after he saw me bag a buck on his first time out with me at age 9. I would have taken him sooner, but I was having trouble with the Ex. #3 was sort of the same way. He started coming out with me at age 4, and got his Hunter Ed out of the way by 8. Then it came time actually do it, and he balked. He was happy just to accompany me to the stand and watch for two more seasons. However, the bug finally hit him this spring, getting ready for turkey season, and he's been begging to shoot the 30-30 at every opportunity.
Some other thoughts:
1) Although our family has eschewed Yute rifles, we are big on Yute loads. In fact, I use H4895 in all my deer loads and load well under the max. As a result, my kids were shooting 30-06 as early as 10 and loving it.
2) Try to find a venue that keeps the shots short. Kids get worked up on their first few deer, and a lot of things can go out the window in a hurry.
3) Shooting offhand was a confidence builder for my sons, but not a requirement for the actual hunt. Make sure they're shooting off a sturdy rest the first time they shoot a deer.