Born and raised in Montana. Hunted elk all my life. Have shot and killed them with 100 grain bullet from a 243, do not recommend it as a primary elk rifle. 30-06 is plenty but bullet and shot placement is most critical. My favorite for the past 20 years has been a Ruger M77 in 7x57 140 grain nosler partition gets the job done. The 7mm08 is the ballistic equivalent of the 7x57 and is easier to find. Light recoil and enough energy to get the job done at reasonable range. One shot kills come from being accurate and competent with whatever you shoot. Just my opinion as to what might suit your needs. Whatever you choose I wish you good luck,
I'm from one state over but have been elk hunting in both Montana and Colorado. .25-06 was my caliber of choice (with premium bullets of course) for those hunts, but I would recommend a little more horsepower.
.30-06, .308, .270Win, .280Rem, 7x57, 7mm-08...any of those "medium" calibers are perfectly adequate. If you're wanting to take those big long quarter-mile shots then I'd say you want a magnum cartridge in any of those medium bores. As others have said, it's best to see what's commonly available in your area or the area that you're hunting.
My personal preference is to carry a rifle that doesn't kick the snot out of my shoulder and I'll just stalk in a little closer to be sure of a clean shot.
Next time I go, I'll be using one of my 6.5x55 rifles.
As for the 770 Remington...
They work. They're inexpensive. The action is a little on the sloppy-built side. Not many accessories available for them. Not the best attempt that Remington ever put forth at building a good-quality "budget" rifle.
Soundguy's recommendation of a Savage would be a better choice.
I'm still a fan of the old "proven" designs like the Remington 700, Ruger 77, Savage 110, Winchester 70, etc. Although I do like the new Ruger American and the Tikka T3.
Don't discount whatever is on the used racks at your local shops either.
If you can overlook some scratches and dings you might find a really sweet shooter for a bargain price.