I have used a .243 for years for whitetail. When I first got the rifle in the 80s my dad had already done some grouping tests with it. He shot 8 to10 5 round groups of different manufacturers factory ammo in ranges from 80 to about 105 grains. The factory loads that my rifle liked the best were Winchester power points 100 grain. I have done that group tests a few times over the years on different lots and loads and right now Im getting the best results with factory Federal Fusion 95 grain.
If you are loading your own Im sure that there are some folks here that can point you in the right direction.
If you are using factory ammo then go find a number of boxes do the group test, let the rifle cool, between groups, and see what the rifle likes. Then use the remainder for practice and buy extra of the brand, grain, lot that shoots the best for that rifle.
Depending on how old your son is you could also get some Hornady Custom lite ammo to test out and let him get used to the rifle. its a managed recoil round,