If you're strictly hunting in woods, you can't beat a good ole 30-30. Ammo is cheap (comparatively) and available. I used a Glenfield (El Cheapo Marlin) for years with great success.
If there's a chance you'll be hunting open fields (or possible a trip out west, or bigger game than a deer), then the 308 would be the better choice. Recoil is minimal, range and stopping power is good.
Personally, I have a Rem. 700 in 30-06, which I've used to take rockchucks (like western groundhogs), deer, caribou, and moose. I reloaded my own (165 grn spitzers) which is a great rounds. A great all-around cartridge.
Better to be over-gunned than under-gunned. A 308 will always fit the bill, whereas the 30-30 may come up short ( you spot a deer in a clearing 200 yds away).