From personal experience go with a 20GA semi auto.
I started my 9 year old boy with a cheap lightweight single shot Academy 410 with some "cool" tactical looking phenolic stock. You can't hit the broad side of a barn with the damn thing. And nothing curbs enthusiasm for kids faster than not being able to hit anything. Also, it kicks like a mule relatively speaking because it's lightweight w break open action. The older and heavier ones kick a little less but not much. And lot of the older ones are dangerous IMO.
When he was 10, I took an older S&W 20 GA semi auto to a local gunsmith. He shortened the barrel, added fiber optic sites, added a choke, shortened the buttstock, and added a kick-eez recoil pad. If I remember correctly it was like $500-600 or something like that. It is heavier than the 410 but he handles it great. There is much less felt recoil too. He's 11 now and knocking doves out of the sky!!!
We use heavy dove loads because little there's not much recoil difference, more & faster load, and they cycle better with younger/lighter kids.
Hope this helps.
ETA - And buy them a proper browning citori or beretta silver pigeon as a Texas A&M graduation present! Ha!
