.308 Winchester
The .308 Winchester is the commercial name for the 7.62 NATO (7.62x51mm) round. The 7.62 NATO was developed to duplicate the performance of the military .30-06 round in a shorter case. It basically does. Winchester released the round to the public in the mid 1950s. With equal barrel lengths, the .308 runs about 100fps slower than the .30-06, with the same bullets.
Commercial .308 ammo is loaded hotter than GI 7.62NATO, and is not recommended for use in GI rifles.
I have several rifles in .308 ranging from Remington bolt actions to Browning lever guns, even some converted milsurp Mauser and Arisaka bolt guns, as well as a couple of "battle rifle" copies (semi auto only) in 7.62 NATO.
The .308 is chambered by someone in every style of action, single shot, bolt, pump, lever, and semi auto.
With only about 100fps difference between them (with factory ammo) there is nothing you can do with a .30-06 that you cannot do as well with a .308. There are very very few shooters who can gain any benefit from the very slight difference in long range trajectory, and there is no animal that walks the earth that can tell the difference when hit. Most folks recommend the .30-06 over the .308 for the largest game, because the slight performance difference increases when using the heaviest .30 caliber bullets, and it is always preferred to use the most powerful round available. But the difference is actually greater on paper than it is in the game field. With proper shot placement, there is no practical difference for hunting.
As to scopes, any good scope is fine. Some of my guns have low power scopes for close range work in brushy woods, some of them have variables for better utility over a wider range of terrain, and some of them still wear just the iron sights. I would stay away from the very cheapest scopes, as you may find them to be less than adequate.
The .308 Winchester is a fine general purpose big game rifle (deer, black bear, etc), and while considered on the lighter end of the scale for elk and moose, it will bring home the game, if you do your part properly. Bullets of 150gr and up are preferred for deer, with 180gr usually being the chosen weight for larger game.
I no longer hunt (age/health reasons), but when I did, my Remington .308 worked just as well as anything could. My Dad's last deer rifle was a Browning BLR in .308 win, and I have it now. Not the tackdriver some bolt guns are, but a fine deer rifle nonetheless.
The .308 is an accurate round, and some very accurate rifles are made in that caliber. It is the choice of most military and police marksmen.
Find whatever modern rifle you like best, chances are it is made in .308 Win.