Several questions to answer this properly...

1. Are you recoil shy? If so, the 308 will be more pleasant than the 30-06. If you don't know if you are recoil shy...spend some time at the range with a variety of calibers.

2. Why type of hunting environment...there's a difference between hunting say Western WA/OR where you will be in heavy thick timber with short shooting distances...or in Eastern Oregon where one is hunting mainly field with long (500+ yards) distances. A different rifle is required for these different environments. Heavier bullets/calibers on quick handling rifles for the thick stuff....flatter/longer shooting rounds for the plains!

3. Choose a rifle you are comfortable shooting is just as important than caliber.

4. While 308 and 30-06 are both great calibers (I have taken Elk, Moose, Deer and black bear with a 308...I have also been successful with Deer, elk, moose, caribu and bear with a 270 and 270 WSM...my personal favorites!)...there are many others one might consider based on their hunting environment....ranging from 270 WIN, 270 WSM, 7-08, 284, 7mm Rem Mag, 308, 30-06, 300 WIN 300 WSM....and on and on and on...options are unlimited. Basically, as long as you do your job...anything 270 or larger will take an elk or deer down...but remember to keep in mind the laws of the state you intend to hunt in...many states require 30 cal minimum for elk hunting.