I started my son out with a youth model spring piston air rifle (a Gamo "Young Hunter") when he was 10 years old. It was light, easy for him to load/cock by himself, and still powerful enough to take a squirrel.
After he took his hunter's safety class, I got him an NEF Sportster in 22LR and a Pardner in 410. The heavy barrel Sportsters are way to heavy for a young shooter, and the lightweight Pardners have a lot of recoil for a little guy when you move up to a 20 or 12 gauge. The single shot NEFs have proven to be excellent choices for him. I found a Pardner with a laminate stock and a recoil pad, and we put a laminate stock on his Sportster because he didn't like the looks of the "plastic" stock. I also like the NEFs because when he's ready to move up to a 22WMR, 17HMR, 20 gauge, 12 gauge, slug gun, or a muzzle loader, we'll have the option of adding the other barrels.
I also picked up a Romanian M1969 Trainer rifle for him that he really likes. They are inexpensive, lightweight, accurate, all wood and steel rifles...but they are ugly. We stripped it down and refinished the stock as a father/son project and he loves that gun too...especially when people at the range ask him what he's shooting!
I just picked up an old Remington 513T because I wanted something for him to shoot without a scope. It has a Lyman aperture sight with a hooded globe on the front. It's not a good hunting rifle for him because of the weight, but it will be great for him to learn how to shoot without a scope at the range.
Every kid is different, and nobody knows yours better than you do. So here's my 2 cents...get them something that fits them, make sure that it's light enough for them, modify it when needed so THEY like the looks of it, and make sure that it's fun for them to shoot it. Trust me...if it sounds like a cannon and kicks like a mule they won't want to shoot it again. Have fun...and most of all...be safe.
Moonlitin