Welcome to Shooters Forum, SkyWest!
Good advice from Tacklebury...I would add a couple of additional pointers.
1) Buy him a bunch of 22LR ammo and set him up for successful shooting, with "fun" targets at reasonable ranges. Shooting lots of 22 is the key to your son learning to enjoy shooting and to shoot well. Let him shoot almost
as much as he wants, or set a limit on how many he can shoot in one session. If you notice him starting to get bored with it, as 7 year-old boys can sometimes do, quickly find a more interesting target, or call it a day. (NOTE: Paper bullseye targets get boring for kids pretty fast. Consider soda cans on a string, metal "spinners", or steel plates that ring out when hit. I once printed out full-color sheets of mythical "bad guys" for my son to shoot at...he loved it!)
2) When stepping up to the 243, and again with the 20 gauge, use some kind of recoil absorbing shooting rest, like a lead sled. Also, make sure he has very good hearing protection, as much of the recoil that kids perceive is actually the muzzle blast from more powerful cartridges. With the bigger cartridges, you want to do the opposite of what you did with the 22LR; only let him shoot a few at a time, especially the first few times he shoots it. I would strongly agree with Tacklebury on using irons for the 22, but if he will eventually hunt with the 243, consider putting a decent quality scope on it.
As far as ammo for the 22, it won't matter much, at first. If you want to take the time to figure out which ammo your gun likes, buy 5 of 6 different types and test them with your gun, under controlled shooting conditions. Your son won't care a whole lot, as long as he can ring the steel from whatever distance you are shooting from! I hope you and your son have a great time learning to shoot, and thank you for your service to our country.