308 or 30-06? Do you want a short action rifle (308Win.), or a standard/long action (30-06) ?
There's really not much difference between the two when it comes to practical ballistics. The '06 has a slight advantage of about 150fps in velocity (+/-), and of course, this translates to a little more downrange energy at the target, and a little flatter trajectory.
In both cases (energy & trajectory) the advantage is VERY little.

Given the usual bullet weights used (150gr., 165gr., and 180gr.) either will perform about as well as the other for a wide variety of big game including deer, elk & moose. Advantage to the '06 if you need to use heavier bullets (200gr. and up).

Personally, I've always liked the shorter action in most bolt action rifles, and have picked the 308Win. For a lever action, barring selection of a used Savage 99, you're pretty much going to be stuck with the Browning BLR - a good rifle for sure, but not my pick. I'd look to a bolt action. Anycase, between these two cartridges, as popular as they both are, just pretty much pick the rifle you want as it will likely be available in either chambering.