Get a good bolt .308 with a 20" ish heavy barrel. It'll be very light in the recoil department, will be short enough to maneuver with very easily in the woods and will also have enough oomph for doing everything you described doing.
Plus it's a decent combo for most mid range shooting at whatever range you may shoot at. I have a Remington 700 SPS Tactical in .308. I hunt with it and can reliably ring steel at 600 yards (as far as my shooting range goes).