Start young kids with a 28 gauge or a 410. The recoil is mild and they can learn to shoot at moving targets. A .30 calibre carbine is also excellent for children when the have moved past a 22 and want to shoot something that makes more noise with little recoil.
The problem with that approach is that 28 and .410 ammo is getting prohibitively expensive. As a high volume shooter of both, reloading is a must.