In my opinion the .308 win/7.62x51 mm is extremely undersized for big game hunting out past 350-400 yards. Don't get me wrong it is a great paper puncher out in the half mile distance, however it simply doesn't have the downrange energy to put a whollup on an elk, mule deer or even a whitetail. I truly believe that if the military had not committed so heavily to the .308 as a "sniper" round, due in large part to the mild recoil generated that it would not even be considered a long range round today. The military has been so concerned about recoil that people now are beginning to believe that even a .223/5.56x45mm is an excellent choice for self defense rifles. I think that a .223 is small for shooting at a little 145 lb whitetail doe, let alone a 220 lb person...
I may be in the minority here but I think that recoil is secondary to downrange terminal ballistics. I don't know any person who was actually injured from the recoil of even heavy magnum rifles. Yes there may be some discomfort for a moment, but the truth is if you simply shoot the rifle a lot, you will get over that little shoulder thump. People seem to be very worried about the ability to get a second shot of as soon as possible. How about making the first one count, and react accordingly to the developing situation?
I think that an appropriate first hunting rifle would be a .300 SAUM, .300 WSM, even the new ruger compact magnums are great for shooting a large mule deer or an elk out near 500 yards. These cartridges do not generate much more recoil than the .308 but the downrange energies are considerably larger.
I've seen a lot of big game animals killed with small caliber varmint rifles and they died just the same as ones shot with a .340 Weatherby. But if you have a choice I would go for a larger cartridge with more punch. Chances are you will be able to see a lot more game fall in their tracks, rather than having to try to follow a minimal blood trail for half a mile.
Please take any "advice" from me with a grain of salt. I realize that the .308 is a very established cartridge and I'm sure that I will always own a rifle chambered in this round. I just feel that a better platform is the AR-10, M-14 types of actual "combat" small arms. All of this is simply in my own humble opinion.
I hope this sparks some good conversation, and no matter what rifle you settle on, just try and get out shooting it as much as possible!