the 243 is a great round. The 25-06 isn't much if any worse than the 243. The 243 is a necked down 308. The 25-06 is a necked down 30-06. If you reload, it's not too bad. any more it seems like 243 ammo is getting just as spendy as 25-06 which I can still find for about $1/round. The 243 with a 90-100 grain bullet is more than capable of killing deer cleanly at 300 yards. 25-06 shoots between 100-120+ bullets. THe bigger 257 bullets will whack a little more than the 100 grain 243 but not too badly. My oldest has been shooting a 243. I'm going to get him on a 30-30 or my 25-06 this year.
If you put 150 grains in the 30-06 it's not too bad. Just stay away from the 180-200 grain bullets
7mm-08 is always in Walmart or Dicks Sporting Goods and is also a perfectly fine deer caliber.
IMO, .270 is a bit much for a younger kid--again, it's a necked down 30-06, but shoots a larger bullet than the 25-06. Bigger the bullet, the harder the recoil.