I own a bunch of 22's.
Here's a short list in no particular order a few guns that are fully up to the task and I'm certain any of us gun guys can appreciate.
Marlin Model 60
Ruger 10/22
Savage FVSR bolt action (I particularly like the one with the 16" fluted barrel)
Henry 22LR lever action.
As for handguns for the older children
Single action 22 of some sort is a timeless classic. Whether Ruger Single Six or their Wrangler or even a Heritage Rough Rider is another alternative.
I taught all 3 of my sons to shoot a handgun using both the heritage and a little Walther P22. They're small and compact. Easy for them to handle and manipulate and are accurate enough to hit paper plates with them. The p22 won't win any accuracy contests but are great for teaching basics and allowing them to grow into better ones.
Another good choice that's been around a little while is either the sig 1911/22 or the GSG 1911/22. Same guns, just different rollmarks and of course the sig name on the side costs a little more but is the same exact gun other than the marking. So, if you would like to get them started right with a 1911, that would be a great start for them.
Whichever you choose, I'm sure it'll be a good one.
There's really so many out there now that could work. There are 22lr AR's. MP5s (which would be kinda cool to get an HK mp5 22 (yes they do make them). So, if you're wanting to get a lot of extra street cred with the older grandkids that are probably getting into video games pretty heavily, those would probably be on your short list to be the coolest pappaw ever. They may not appreciate the single and lever actions as much as we do. Though, we both know they're well up to the task.
None of the ones I listed are over $450'ish at the most except maybe the ruger single six.
My opinion is to get a good base gun and spend the rest on ammo and use that money shooting it with them. Those are what they're going to remember most. Sure, they'll remember the gun too. But they will appreciate the time with gramps more than the gun.
Hand each one of them a 100 round box or one of the 333 pack of shells and watch their eyes light up. It's not going to be all about the gun. It's going to be all the good times they're about to make with you shooting until they run out.
I appreciate you nitesite. Much more than you may realize.