There is no one gun or cartridge that works for everything. That bears stating, as usual. The .30-06 Springfield is one of the best all-around cartridges.
It can be used for anything from deer to brown bear, even though it's at its best against deer or elk. It's a bit much for varmints like coyote, but for cougar it would be a decent choice. When it comes to "all-purpose," I prefer to give three guns, rather than just one: 1) .22LR, for plinking and small game.
2) 12 gauge shotgun, for anything from small game on up to the biggest of the big (with slugs, of course), but only at relatively short range. 3) a centerfire rifle that's appropriate for the game you wish to hunt. .243 Winchester is fine for whitetail and mule deer, as well as pronghorn, coyote, and in the right hands even wild hogs.
.30-30 Winchester is a short-range cartridge, but if you're only hunting in close cover for deer, it's a fine choice. .308 Winchester is a good all-around cartridge, but it has less power than .30-06 Springfield for the same price. .270 Winchester is a fine choice for anything up to elk, and can be used for moose if nothing else is available.
Incidentally, moose are much tougher than elk. With any cartridge, you pick the bullet for the game. A soft point or ballistic tip bullet is fine for whitetail, but next to useless for moose.
A monolithic bullet makes a good choice for very large game, such as grizzly or moose, as do bullets such as the Nosler Partition and Swift A-Frame. So with all that said, let me sum up: if you do not already have a .22LR and a 12 gauge, get them. And then, get a .30-06 Springfield.
That's if you ONLY want to look at .30 caliber choices. There are a lot of other choices, and you should consider them if you will never be going after moose or grizzly.
I rarely recommend 30/30;it has a limited range/energy profile that can be easily surpassed by more "modern" cartridges. Consider the 308 (7.62x51) as 30-06 Lite,about 170 FPS slower than 30-06 and not quite as able to handle 180 and heavier grain bullets. I've used 30-06 since I was 13,pretty much 40 years,for deer,elk,bear using 165 grain Rem Core Lokt bullets.
Now that I reload I use better quality bullets,and many ammo companies provide quality bullets in factory loads. So, with 308 or 30-06 you can select the bullet for performance by weights and designs-and bullet design is more important than actual caliber for most centerfire cartridges from 243 up for deer,but for elk/moose minimum I suggest is 270-so going with 30-06 or 308 is fine. There are still Accelerator loads available in either cartridge (or reloadable) that use a 55 grain 223/sabot for varmints too.
Personally,since I grew up with 30-06 it has always been my choice,but nothing wrong with 308.
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