In my experience lever actions seem a bit less accurate than turn-bolt rifles. I have lever actions in .243 Win, .30-30 Win, and .44 Magnum, and tend to use them in more brushy terrain where shots are closer.
I have bolt actions in .350 Remington Magnum, .270 Winchester, .25-06 Remington, and .22 LR. I am comfortable with the .350 out to 400 yards, and the .270/.25-06 a bit farther. I qualified with my .25-06 on a 1,000 yard course, but would not shoot anything but rodents at that range (though best shot was 403 yards on a p-dog).
Both .308 and .30-06 are solid choices. I've used a .308 out to 800 yards on a USMC rifle range and it worked as advertised.