30-06 recoil will get old. 308 is plenty for deer or moose. You can load it down to 125 grains all the way up to 220 grains.
I'm not knocking the 30-06, but it's really overkill for deer or moose. Less recoil makes you more accurate and shot placement is more important than the small ballistic difference between the 308 and 30-06.
Trust me, the 308 is an amazing hunting cartridge. It's good for anything on North America minus the grizzly.