Well the first thing you should have said is what type of terrain you expect to hunt in. it would helpful ...I have hunted deer from age 14 and now at 72 last deer in Ga. this Dec. originally from NJ (not by choice) we basically hunted in the northeast . that being said the possibly of hunting in PA was the reason I did not get a Rem 7400 in 308 ( semi's not allowed for big game in Pa.. Since I always used pump shotguns I'm satisfied with the 7600...
I think that the 308 or 30-06 is on he top of the list.. I have 2 friends and not knowing to either of us we all settled on the Rem 7600 pumps in 308 or 30-06. Two of us are lefties.. why not check out some of the best deer hunters in the northeast... google "Benoit Brothers" you will notice that they all hunt with pump rifles.... For Maine deer / bear you leave the small bores at home ..I hunted in some of the same areas that they hunted and the pump is well suited a quick second shot.
When it is going to be your first and maybe only rifle for quite a while you should get a caliber that can knock down just about anything short of a grizzly charge and a well placed shot could probably handle that too.
It's also nice to have a 308 (7.62 NATO ) in the closet.....
I also have a Rem 700 LH 7mm mag but that is for long range western trips.
also had a Savage 110 LH in 30-06 great light bolt action rifle (stolen)
when I was young (1962) I loaded my 30-06 150 grain pushing 3000fps. well now Hornaday has a 308 150gr light mag load pushing the same FPS as my old 30-06 HOT load..... Good Luck on your choice..