Deer and largish game?
Chalk up another vote for a cheap surplus rifle that you can buy cheap surplus rounds for. Best candidate really is a Mosin, I nearly got into deer hunting and did my homework. A tin of 400 surplus rounds can be had for around 100$ which pretty much as cheap as you'll ever find ammo in that class of firepower. Make sure the crowning on the barrel is still in good shape (Meatshield infantry doesn't care about keeping their gun in shape, the way the included cleaning rod was usually used damaged many a Mosin barrel's crowning to hell).
Rabbits and small game?
FUCK .22
I think it was the Obama scare that caused a bunch of right wing nutters to stockpile ammo, but whatever happened the price of all ammunition skyrocketed if you didn't notice. .22 is about twice as expensive as it used to be,
and that's when it's on sale.Look into the laws regarding airguns for hunting small game. Much cheaper alternative in the long run, or at least until ammo prices drop (which they probably never will).
If you do get a .22, I like and highly recommend Rugers. You couldn't miss with one if you tried. A nice one can be had for as low as 300$ if I recall, and I think you could get one with a bull or fluted barrel for not much more. Wasn't mine, so I didn't have to clean it
but I'm told they're a little complicated to disassemble and reassemble.