$750
I would go with a Thompson Center Encore in 223. I would buy one of the Nikon Monarch 3 X 9's that many sellers have on closeout right now. With a little shopping you should be able to get into this outfit for $750. My reasons for picking this outfit are simple:
1. Accurate right out of the box.
2. 223 is a great round for what you described. accurate, minimal recoil, large selection of ammo choices, and inexpensive ammo, so you can do alot of cheap shooting.
3. Encore is a great choice for additional barrel selections at a good price. If you decide to do other types of hunting, just buy another barrel.
4. The Nikon would make a good quality scope, and still give you a decent price.
Tom.