Maverick 88.
Remington 870, Winchester SXP, Mossberg 500/590 are all awesome guns, but they cost more. From what I've heard the turkish guns are good, but.... for the price, I've never heard a complaint about a Maverick 88 (which is very very similar to a Mossberg 500).
The Mav 88 with 18.5" barrel will do everything you need it to do, mediocre but reliable. And for under $300.
As for caliber, just go 12 gauge. Keep it simple. Vast and cheap. 8 Pellet buck recoils about the same as full power 20 gauge anyway, but the 12 gauge is still delivering more lead.