+1 on the separate .22 & shotgun.

If skeet shooting or bird hunting are an option, I'd definitely go with a 20ga over the .410. The 20ga has more recoil but it will be much more effective. Ammunition will also be more widely available for the 20ga. In normal times down here, anywhere that carries ammunition probably has 20ga & 12ga. I wouldn't go with a 16ga or 12ga. They are a BIG step up in recoil (I have never shot one with reduced recoil loads). What type of country are you going to be hunting? For upland hunting (in brush/woods for squirrel, grouse, deer, etc.) you will probably want to get a shorter barrel.

For a 10 year old, I'd start with the .22. Especially on a tight budget, you could let the shotgun wait for a couple of years. I don't remember how old I was when I started shooting a .22 with my granddad but i didn't start hunting with a 20ga side by side until I was a teenager.

P.S. Do NOT shoot nearly straight up (butt of gun pushing down on top of shoulder) with a 12ga if you end up with one in the future. BTDT