My son started with a .22, then played with a .223, 44/40 and .243 a bit. By the time he was 11 he was shooting .257 Weatherby, 20 gauge and 50 cal muzzleloaders with 3 pellet loads. The last one may have been pushing it a bit, but the others were/are mild choices and what he hunted with. When he grew into a 12 gauge my daughters got the 20 gauge pump.
My daughter went .22, .223, .243, 7/08 at the same approximate ages. Looking back on it, we could have likely skipped the .243. I don't think that there's anyone that could handle a .243 but not be able to shoot a .270 or 7/08 with starting loads at regular bullet weights, or light bullet handloads.