Just Learning,
Welcome aboard to the wonderful world of teaching your own kids to shoot. We just started this year and it has been so gratifying to watch my girls (the two oldest anyway, ages 12 and 11) learn to respect and use a firearm. I hope you get the same thrill and sense of accomplishment out of it that I have.
My girls first dry-fired and then live-fired a scoped .22 bolt-action Glenfield Model 25. It's nothing fancy, but it is a bolt action and is scoped, just like the deer rifles they will eventually be using. Once they got comfortable with the .22 I introduced them to the deer rifle (only the 12-year-old gets to qualify this year; each one will get special attention their first year of hunting). I chose the 6.5 x 55 caliber Swedish mauser for several reasons, not the least of which is it's a darn nice caliber that has plenty of punch for deer-sized game without excessive blast and kick. Sidebar: it is very sweet-shooting and has me itching to hunt with it now, too!
I don't think the actual make/model of .22 will matter a whole lot as long as it can shoot fairly accurately, is mechanically sound, and similates to some degree what the child will be moving up to some day for hunting larger game. The rules of shooting, good habits instilled, and positive reinforcement from you will outweigh by far whatever type of gun it happens to be.
Good luck with your shooting lessons and first hunts!