First off, I would definitely stay away from the magnums. For the circumstance you have described, they are not needed. While others have given good advice on the various cartridges, my choice would be the .308 Winchester hands down. The 30-06 would be my second choice and the 7mm-08 or 7x57 Mauser my third choice. That last one (7x57)is a darn good round but is underloaded by the factories. It can be souped up if you're a relaoder to what the 7mm-08 can do and go with heavier bullets to boot. A great round and I like it.
But, let's go back to the .308 and why I rated it first. One, is is just about as powerful as a 30-06 and in some cases more powerful.
What did he say? Yup. I have chronogrphed several brands of ammo in both .308 and 30-06 using 180 gr. bullets and the .308 was faster. It is my thought that the liability lawyers have suggested that the factories quietly down load the good old 06 just a bit more than they used to due to the old weak guns still in use. Let's just say that one brand I tested had the .308 going almost 50 FPS faster than the 30-06 in the same brand.
My personal choice for the .308 as an all round load would be with a 165 gr. bullet. I took a fairly large Mule deer at 250 yards with a 165 gr. Speer Hot-core loaded to 2550 FPS from an 18.5" barreled carbine. (Ruger 77 RSI) The same load delivers 2610 FPS from a 22" barreled Winchester M70, for whatever that's worth.
The point is, although I could load up faster loads, this one does the job. I forget what load I used in a 20" barreled .308, but the 150 gr. bullet pushed by a hot load using H-335 took down a 250 plus pound Mule deer at 427 paces, witnessed by two hunting partners. The 150 gr. is fine for deer, but mangles too much eating meat IMHO. Most 180 gr. spitzer bullets may be a bit too tough, but they should work OK. I feel the 165 gr. bullet is an excellent compromise, and I wouldn't worry about that load on elk out to about 200 yards.
I must like the .308 quite a bit as at last count, I have seven sporting rifles and one target rifle chambered to the round.
One of the sporters only weighs 5 pounds, loaded, scoped and with a sling. That one only has a 19" barrel. Recoil is not as bad as you might think with that rifle, but holding it steady is a bit of something else.
Look for something like a short action Winchester M70, Ruger 77, Savage 110, Remington M7 or 700 or my favorite, a Ruger 77 RSI in .308 and have fun. Every one of those rifles should do the job for you with ease, except the RSI. They can be a bit fussy about ammo, but once you find the key, they'll be good shooters too.
Paul B.