find a local hog or chicken farm and see if you can sit around their "dead pile" or compost heap. try to learn what they do and what time they do it, so you can ambush them.
I'd get a .223, ammo is cheaper than the other calibers and it is plenty capable.
After you try to skin one you will change your mind about keeping pelts.
Thats just my guess.
If you want access to land offer to help maintain it rather than just hunt it. What do you hunt now? Are you good at it, safe?
I thought about the AR but the entry price is a lot higher. The ammo I have seen is $105 for 7.62x39 500 rounds and $130 for.223 500 rounds. So pretty close but AK a bit cheaper and I don't worry about firing cheap ammo on the AK