Your instinct for a .308 is a good one. Just a touch less than a 30-06 (maybe 50 yards), but a very smooth shooting rifle. I find it more pleasant to shoot than a 30-06 and much more pleasant to shoot than some of the more powerful cartridges mentioned in the good advice above. The 7mm-08 is a nice round (a necked down .308) but is hard to find in heavier bullet weights.

There are some fine recommendations from others here, but the .308 does not punish you for shooting it. Some of the larger and magnum calibers would discourage me from practicing enough. I had a 30-06 (my second choice) and sold it, shot a 300 Win Mag once (once was enough) but...I'll admit I am not a power shooter like others may be. I use 165s for everything and have but a single hunting rifle but know it well.

If you get a bullet drop compensating scope with range dots on it, you can zero the scope for elk/deer loads. Then, buy some subsonic .308 rounds (they make them) and see which range dot puts them on target at shorter ranges. My Nikon is the 5th one down at 50 yards.

That way I can hunt with a big game round in the chamber but can rack a round and shoot grouse or small game with a subsonic round without blowing up the game using the correct aiming dot in the reticle.

It is a very versatile setup. The third round is another high powered round for follow up shots on big game (which I have never needed, but need to be prepared for).

I live in the Colorado Rockies hunting deer, elk, and pronghorn, and my second choice would be a 30-06, as so many others have mentioned. But, I have never found subsonic rounds for them (though they probably exist).

Best wishes as you decide.