Greetings. I'm glad to see yourlittle burgsurvivedmy 4 year stintacross the rivermany years ago.
Start by reading this:
http://www.huntingnet.com/forum/tm.aspx?m=2015336
and then all8 pages of this:
http://www.huntingnet.com/forum/tm.aspx?m=1856380.
You'll find you're on the right track. You're also about to get hundreds of thousands of replies from zealous defenders of 1 cartridge or another, as well as every model of rifle ever produced: be prepared to take them all with a grain o' salt. The .308 & .30-06 will comfortably do what you'd like to do, as will several others. And the Stevens 200, despite its low price and blue collar pedigree, will put the bullets awfully close to where you want them to go.
In your price range, you may do well to take a look at Howa rifles, too, since you'dget a better trigger & stock. And please, PLEASE, make room in your budget for at least $100 worth of scope: you'll never hit what you can't see.
Good Luck,
FC