It's difficult to recommend a good hunting rifle without having a rough budget number to work with. Recommending an expensive cooper would be pointless if you have $700 to drop on a rifle and sight. That being said, here are some popular makes/models:
Savage Axis - Can usually be found for around $400
Ruger American - Also about a $400 rifle, pretty much the same as the Axis in my opinion.
Weatherby Vanguard S2 - About $550, probably the best rifle you'll find for under $1000 imo.
Howa 1500 - exact same gun as the Vanguard, however I've only ever seen them with a Hogue overwrap on the stock for about $700
The prices go up from there, most of these rifles will go for $800-$1100
Browning A or X Bolt
Winchester M70
Tikka T3
CZ 550
Also keep in mind those are all prices sans scope, so add another $200-$500 for a decent to great piece of glass. I will personally vouch for the $200 Redfield Revenge, I put one on the .300WinMag Vanguard S2 I bought and it's held 0 after about 100-120 rounds and managed to tag out with the combo this year.
Caliber wise, I'd go with a 30 cal something (.308, .30-06, etc) good enough to take anything on the North American continent. However .270 and 7mm-08 are also good hunting rounds.
And as stated before, you're milsurps are capable of taking game as well. Many an everything has been harvested with a 303 brit, 7.62x54/39 and pretty much anything else that will fire a projectile of some sort. I know if I had an M44 that I could/'wouldn't feel guilty about' scoping i'd hunt with 7.62x54r anyday.