My personal choice would be a Savage bolt action in something along the lines of .22-250 or .223, with a 26" heavy barrel, and a decent 4-16x__ magnification scope with a bullet drop compensating reticle. I like Savage actions, 26" of barrel gives you a good boost on velocity once you get out there, and .223 or .22-250 won't destroy the pelt too badly. A good scope is key and a 4-16 gives you plenty of flexibility range wise.
An AR-15 will certainly do you well though, and if you want to hunt deer, you can just swap out the upper to something bigger. You can get a 24" varmint barrel and throw a good scope on it, but it will cost a good bit more than a bolt action.
Edit: Under $400, I say go with a Savage bolt action in .243, .270, or .30-06. Dicks Sporting Goods has had them as low as $350 including glass on sale. A decent AR will be well out of your range. Also, .30-06 ammo is everywhere and can be found cheap which is a nice plus.