Unless you are shooting across fields you don't need a ton of power. In the woods, the old .30-30 in the 1895 gun has plenty of juice. And the lever action is nice if you need to take a second shot. Accuracy isn't as good in the lever gun, but again, in the woods you won't be making 200 yard shots.
You'll find plenty of guys grossly overpowering their guns. 7mm and .300 mags are great at 300+ yards, but it's just a pointless beating at closer distances. Don't succumb to the temptation to get more power since magnum bolt guns cost the same as standard calibers. If you go with a bolt gun, .308 is a mid powered cartridge that will easily get out out to 200 yards. 7mm-08 is my personal favorite, because it shoots a little flatter and kicks a little less. It's not as common though so you might need to check ammo supplies locally.
A great gun you might consider if your budget allows is the Browning BLR. It's a lever gun that locks like a bolt action, so it can fire the popular deer calibers like .270 and has accuracy like most basic bolt guns. Nearly any maker's mainstream bolt gun will work fine as well, and most are cheaper than the BLR.
I wouldn't recommend junk guns like Nagants because the triggers are terrible and they are cumbersome to carry. SKS rifles aren't much better, although the auto action comes in handy about 1% of the time. You need your first rifle to be an easy shooters gun. If you go ahead and get what you need instead of making compromises you'll come out ahead in the long run.