two quick suggestions.

1. Instead of the 22 rifle, consider a rifle chambered in 22WMR (aka 22 magnum). This is a slightly more powerful version of the 22, but still is very family friendly. the ammo is also relatively inexpensive, but not as cheap as 22's. the 22WMR is an excellent general purpose cartridge (bullet) for survival purposes, because it will bring down larger game than the 22. technicaly you should not fire a 22WMR at a deer (it's illegal during hunting season). practically, if your family's lives were at stake then you could kill a small white tail deer with a close range shot (say 50 yards) at the head with a 22WMR. so a rifle chambered in this round is well worth contemplating.

2. I heartily support the suggestion of a 20-gage shotgum for general purpose survival and defensive purposes. in fact if you had to choose one long gun for all family purposes in the wilderness, then the 20 gage would be a mighty fine choice.

the 22WMR rifle would be fine for the kids and you. both you and the hubby could handle the 20-gage and be very effective for keeping everybody safe and putting food into the pot.

finally, if these options worry you at all - consider a high-caliber air gun. You can get air guns in 22-caliber and 25-caliber. these air guns are nothing to sneeze at - they will definitely kill small ground animals (rabbits, marmots) and all manner of birds. they are pretty accurate out to 50-75 yards. but the good air guns in these calibers are not cheap (maybe $500-$800) so you are not saving money. you can get pretty quiet versions though, so you can use them in your own yard for target practice.

good luck,
Pete2