Not knowing your grandchildrens' ages or physical condition I'd say .308, 7mm-08, or .25-06. At 11 I started with a .243 and quickly moved to a .270. I'd say that was a bit much then, but 5 years, 100lbs, and a muzzleloader later it's almost indistinguishable from a .22, which may be something to keep in mind.
The three I mentioned above can reasonably kill anything you can shoot in North America which doesn't require a magnum caliber, but may not be the absolute best first firearm (I'd pick a .243 for that).