Any of those three calibers will do in any critter in North America. If you know you're going to go after elk with it, you've got the right idea about what range of calibers to go after.
IMHO, you need to pick a rifle. As long as it's in one of the calibers you're liking, you're GTG. Then base, rings, scope, and sling, and it's done. If you go used, you may find a package all ready to go that is a lot less than new. Or you can get a Savage 11/110 and a Burris Fullfield, Nikon Prostaff, or Redfield Revolution, and go to shooting and hunting.
Match the bullet you use to the game you're after - light, frangible bullets for varmints, medium-for-caliber softpoints for deer and antelope, heavier solid copper hollow point and bonded core softpoint bullets for elk, bear, moose, etc. That's probably got as much or more bearing on your eventual success as the caliber, rifle, or scope.