Many elk to be found in different areas across the west. Some places require non-residents to put in for the "draw" well in advance. Wyoming wants the applications in January for example. Be prepared to pay for the non-resident tag up front too. If not drawn, they'll send back everything minus a small processing fee.
Some areas are relatively easy to hunt. Some are quite difficult because of high altitude and rugged terrain. My Wyoming bull was taken at about 10,000' ASL, a mile from camp and ten miles from the nearest road. If you're hunting the rugged/high country, being in shape is vital. Then, give yourself a few days to acclimate to the altitude and lack of oxygen.
Without pack animals I couldn't have gotten him out of there. First day my partner/mentor and I skinned and quartered up the elk and got the parts back to camp. Next day we spent all day getting the meat and head down to civilization, then getting us back to camp. A big lesson learned for this mule deer hunter is that bull elk are BIG. Think 700+ pounds. Many elk taken on public land though are cows, spikes & raghorns at nowhere near that size. Still, they'll be bigger than a big deer. Have a plan for dealing with that. Any bull on public land is a trophy. Particularly on a Do-It-Yourself hunt.
Elk can be hunted at long range, but they may also appear at more modest ranges. Be prepared for that 30 yard snap shot. Your .300 RUM is more than enough gun for the job.
Back to where??? Washington and Oregon have big-bodied Roosevelt elk. Monsters in body size and usually heavy antlered too. The Rocky Mountain elk are also present in Washington and Oregon but are in big numbers throughout the western states. It might be difficult to pick a state. I've hunted them here in Washington where I live, and in Wyoming when I got an invite from a friend there. If you consider Wyoming, beware that a guide is required for non-residents to hunt the designated Wilderness areas.
In addition to the good input I'm sure you'll receive here, you might want to check out Eastmans Hunting Journal. As a subscriber you'll be able to access volumes of research material on hunting "out west." Many western states have vast areas of public land free for you to hunt. Figuring out where can be a daunting task. For instance here in Washington there are many areas open to elk hunting, that actually hold very few elk.
Elk hunting is one heck of an experience. Particularly if you can arrange to hunt the rut. Having a ticked off 700 pound bull screaming at you, looking for a fight, is simply awesome.
Best of Luck! Guy