If your small game includes coyotes or bobcat sized animals you might want to think of the smaller centerfire cartridges, such as the .204, .222, or .223.
Any .22lr will do for bunnies and squirrels. There are a bunch of used ones to choose from. My brother and I shared a Remington Field master as our first rifle. I recommend them highly. But any rust free used .22 that feels good to you will be fine.
I've used a 20 guage shotgun on rabbits. They work great with bird shot, but I pretty much decided that a .22 would do just as well.