Alsatian

You'll hear different advice from different people. I would advise buying a rifle chambered in .30-06. This cartridge is suitable for all big game in the lower 48 states. You can find it chambered in almost every rifle you would want to buy. You can buy ammunition for it anywhere. I have hunted elk 5 times in SW Colorado and taken 3 elk with my .30-06. I see the .30-06 used a lot in the elk hunting mountains.

I would suggest you buy as cheap a rifle as you can find. You can consider buying used -- either from a private owner or from a pawn shop (be advised, however, that pawn shops sometimes put an excessively high price on used guns). As an example, about 10 years ago I bought a Remington model 300 ADL in synthetic stock for $280 at a grand opening of a Bass Pro Shop. While that rifle is not pretty, it shoots very accurately.

If you have money to put a good scope on it right away do that. But if you can only afford a cheap scope now . . . save your money, accumulate cash until you can afford a truly good scope. Your rifle may have open sights to begin with. People have killed a lot of game over the years with open sights. You can too. You can decide what constitutes a "truly good scope." I put a $500 Leupold VX-III 3.5-10x 40 mm objective duplex reticle scope on my $280 ADL.

Over time you may buy other rifles, nicer, more expensive, more beautiful walnut stocked rifles. This initial cheap rifle can serve as a back-up. That's why I got my ADL, as a backup that I wouldn't feel uneasy about taking out in foul weather or to fill the place of my go-to rifle if I drop it out of the truck on the concrete at the hotel at an out-of-state hunt. Note that though the back-up rifle can be cheap and homely in appearance (which do not generally reduce the accuracy of a rifle), if you are going to hunt with it -- possibly during an expensive out-of-state hunt -- you will want to be able to depend on the sighting system. You don't want a cheap rifle scope fogging up on you and causing you to not get a shot at the only bull elk you see on your expensive out-of-state hunt.