While my preference is a .30/06, the .308 is a great big game cartridge, the ballistic equal of the .30/06 with bullet weights up to 168 grs, and some will say up to 180 grs. Provided its chambered in an appropriate rifle, there's little point to switching to anything else if you already have one . If your rifle is an M-14 style gas gun, it can be a good hunting rifle if you're young and tough enough to pack it around. If your rifle is a heavy barrel varmint or target rifle, that is topped with a large, high magnification, target scope, maybe you should consider a lighter - smaller outfit. The nice thing about a .308 is that it can be chambered in a small rifle like a Remington M-7 for instance or one of the small Ruger carbines, and there's no flies on a Remington 760 (7600 or 76 or whatever they call em these days). Or maybe your rifle is a handed down Savage 99 or Winchester 88, both of which are dandy hunting rifles.
For medium sized big game, a 165 gr lead core bullet is a good choice, as is a 150 gr TSX. I've been shooting heavier bullets in my .30/06, but remembering how much I liked 165 gr Hornady SPBTs backed with a case full of H-4831, I may have to revisit them.
Of course if you're just looking for an excuse to buy a new rifle and try a new cartridge, don't let any of the above dissuade you. There are plenty of cartridge choices from the quarter bores to the .460 Weatherby, and an equally bewildering selection of rifles to choose from.