Under $100 gun
I usually just read these threads, but I'm going to reply this time.
I have hunted with a single shot shotgun and we used to try to get more than one bird out of a covy, but it's hard to do. If budget is too tight you might not have enough to practice.
I think I would go to a pawn shop and get a Marlin M60 .22. Very accurate with bulk ammo. Get some practice, then give up sodas, coffee shops, whatever you can give up, and save for a good used S&W M10. Wally World has some cheap ammo for .38.
Don't buy junk!