First things first, get what you want for hunting in the area you will be hunting for the foreseeable future. Not sure what part of NY you are in, but there are some really nice whitetail up that way and I'm sure black bear. Get what you are comfortable with. If you like .308 see if a friend has one you could try. It is a great round for hunting in your neck of the woods and I would guess (never hunted Elk) it would take one at reasonable distances.
And for the 1,000 yd itch see if there is a PRS range within driving distance. Most have steel up to 1,200 or so yards and both in my area have two or three range days a year as well as classes on long range shooting. A .308 may be more challenging than say a hotrod 6, but still very doable (white feather made a career out of it.) and you can probably do it all with one gun. I would lean more toward the hunting to start since most competitions take 100-200 rounds per match.
No matter what do not over think it! Your day is full of work, stress and the usual grind, you want this part of it to be fun! I will say out of all the rifles I own some of the best times are when my brother comes down and we burn 500 rounds of .22 at fly targets!
