I'm actually surprised at how many folks here don't mind picking lead shot out of their meat. I haven't used a shotgun on rabbits and such in many years.
They're fine for feathered game, and are oft-times the only legal way to hunt them, but not on furred animals.
.22 LR ammo is inexpensive to shoot, lightweight, and very effective on small game. Head shots cause no meat loss on small critters.
There are so many makes and models of rifles chambered for it that it's hard to single one out to recommend. The newest firearm I have that's chambered in .22 LR is 29 years old, but that old Marlin/Glenfield model 60 with the ugly squirrel on the stock has accounted for more than it's share of small game. It's still in fine working shape, and still hits where I aim it.
One .22 pistol I have was made in 1941, and one old Winchester rifle in the late 1930's. Both still function perfectly, and still shoot accurately.
Which goes to show just how hard it is to wear out a decent made rimfire rifle or pistol. With a bit of care, they seem to last forever.
It's jus hard to beat a good .22 LR for small game.