I'd have to go with a .308. Reasons (from guns I own...)
'06 vs 308 - 308 Milsurp ammo is cheap, short action, .308 IS more accurate but they're close enough that normal hunting it won't matter. Power is virtually the same. Guess it's a cost issue.
375HH vs 308: The H&H has tradition, I love hearing the asparagus sized rounds thunk into the chamber. I have a very high recoil tolerance and find the HH fun to shoot with 300gr SBT's at 2650fps. But: It uses lots of powder, bullets are $$, and most people can't shoot it well enough to hit a coyote at 300 yards (which it is very capable of doing)
.340 Weatherby vs .308: If you tolerate recoil w/o flinching, and you reload the .340 has unmatched ballistics. Basicly a .270 with the punch of a 375HH. 340 has a very sharp recoil punch and concussive muzzle blast. Prone shooting from a bipod is punishing, even for me. Watch the scope etc. shooting prone or she'll move your eyebrow to the other side of your head. I find the .308 easier to practice with and the 340 forces a recoil oriented technique vs an accuracy oriented technique, hence the .340 is mostly a safe queen.
223 / ar-15 vs 308: Love the AR, cheap ammo, accurate, great for plinking and competition, but it's not a moose gun, period.
243 vs 308: I deer hunted with a .243, 95 gr Noslers are deadly for an open shot / good placement, but on a moose it might limit my ability to shoot if presentation was not perfect.
All of my lever guns (25-20 to 45-70) - not really suited for long range.
7-08 vs 308: 7-08 is a better target gun as the recoil is less, but the cheap milsurp ammo and xtra bullet weight of 30cal win for the moose.
270 vs 308: Love my 270. This one is tough. Cheap practice ammo w/o reloading (I'm sometimes lazy...) wins for .308. Otherwise we're splitting hairs, .270 is a bit better for ballistics but the pillar bedded .308 M700 is just so accurate...
.257 weatherby vs .308: I used to coyote hunt extensively with this gun. Loved the mark 5, but I shot out the barrel. It did amazing things to small animals at very long ranges. I couldn't sell pelts from this gun but the price had dropped to $5 in the early 80's and it wasn't worth skinning them. The .308 is more accurate and the barrel will last long enough to find an accurate load.
6.5 Jap / 7mm mauser / 8mm Mauser / 22-250 / etc. have their place but a .308 is more versatile.
Personally I've done very poorly as selecting 1 gun for everything, but get a .308 and it'll get you started.