The .308 is a great round that can be used for just about any rifle game. Just don't get suckered into the ultra accuracy/hyper fast shooting crowd. I've found that the most accurate bullets are those designed for varmints. Those light accurate bits of metal explode on contact and give spectacular kills on game under 40 pounds (like the coyote). Don't feel compeled to use the same bullet on larger game like deer, hogs and bear. Unless you're taking head shots you're just going to wound them. Varmint bullets are designed to open up on contact and deliver as much hydrostatic shock as possible. Doing that requires a trade off in penetration. You don't need much penetration to kill a varmint but you doo need penetration to kill a larger animal. On a deer, hog or bear you need a bullet to penetrate to the vital organs and destroy them by breaking bone and ripping through a foot or more of meat and hide. A varmint bullet won't do it.
the Winchester Failsafe and Remington Core Lokt in .308 will kill a moose. The Winchester Partition gold, Silver tip, Remington Scirocco and Accutip will take care of the biggest black bear, deer and hogs your going to shoot at. The Win Power Point plus is great for the smaller stuff.
Anything better than that is going to require you to handload.