I grew up mainly hunting open fields in SC, with ranges up to around 400 yards. I've shot both a .30-06 and a 7MM Rem. Mag with great success. They are great, but I always stayed in nice stands out in the open due to not wanting to beat up the guns (both very high end Sako's), now in the safe most of the time.
These days I have an inexpensive Remington 770 .270 that I won at a DU banquet. Full synthetic stock with black parkerized finished metal. I put a decent Bushnell scope on it, did a trigger job, and can easily group within a quarter at 200 yards. Plenty of aftermarket stocks available if the factory one is too big, so I would highly recommend this option.
My other go to gun is an old Marlin 30-30, which I greatly prefer in the woods for it's compact size. If you are starting out in the woods, this would be my top choice.
Regardless, the one thing I would be sure of is that whatever you choose has a good trigger, either from the factory or modified by a good gunsmith. I am amazed at the trashy pull quality of the average out of the box trigger these days. I am not a fan of a super light trigger, but do like them to be around 4lbs or slightly less, and absolutely smooth. This makes for a much safer gun (because discharge is predictable) and also much more accurate because you are less likely to pull off of target. A good gunsmith can get you set up for under 100 bucks.