I start my girls out with a 22LR, but they started at ages 4 and 5. When my oldest hunting/shooting daughter turns 10, we will probably get her a 20 ga for deer and rabbits.

For a 10 year old, I still think the 22 is best. Really for anyone the 22 is best. You can shoot hundreds of 22 LR and really solidly learn shooting before you introduce recoil. After that, I stepped up to a 30'06, but like most, it was because that was what we had. I also dabbled in shotgunning with a light single shot 12 ga. . .ouch. . .but my dad shot it well so I figured I would soon be able to shoot it well. I never gave up and finally grew into it! Probably a 243 or 308 would have been a much easier step to make. I killed a lot of deer with that 30'06 though and it didn't have a kid stock either. If you child is smaller though, you will need a shorter LOP. Don't get to light of a rifle. A good 9lb gun or so really helps a new shooter with recoil and shaking with a good field rest. Offhand is sacrificed some, but new shooter shouldn't be shooting offhand.