I'd say get a BB gun like a Daisy first (red rider or a buck or a model 25) then graduate to a Crossman 760 pellet/BB repeater, around 9 or 10yro move them into a .22 cal of their own, have them learn safe shooting, wear shooting safety eyegear, safety infractions they lose privlages.
I had a few single shot .22's they got to try but they never really seemd too compatable (marlin 10, itcha 49, Stevens 30, Stevens 73Y, H&R handi rifle) the kids (2girls) were very excited when they went in the gun shop with Dad (to buy ammo) and spyd a youth model Cricket sitting on the rack, It went home with us that day, it was a blue/black one, they used to argue over who got to shoot it, later on we bought another Cricket (pink/stainless).
its a great littel .22 for the money, later on you can get a adult stock for them.
for a first .22 having it manual cock bolt action is the best safety feature from a parents point of view, the marlin 25 scares the bejezzus outta me, cock the knob when your ready to take your shot, easy to uncock and unload as well.
Id taught all 4 my kids to shoot and they are pretty good shots, as kids you still have to keep an eye on them, even the 16yro bears some watching & reminding ones in awhile.
Its best they learn early on just what these firearms can do wither a BB gun like a red rider (painfull welt if accidently shot) or a .22 holes through trees and 2x4's.