Carry two good knives with you. If you get an elk, you will want to field dress it promptly. Two knives because sometimes knives break on elk. If it is warmish, you might want to skin and quarter the meat right away. Have a plan for carrying out the elk on your back in 50 LBS loads -- four or five trips. If you have horses, mules, or a motor vehicle that you can use to retrieve your meat you are lucky.
i have heard it said that you don't want to move slow when looking for elk, as that prevents covering a lot of ground, and you need to cover a lot of ground to find the elk. If you make noise, that isn't necessarily bad: elk make noise too. When you find fresh sign, then slow down. Use glassing with binoculars early and late to find animals while saving your legs.
Elk are not just anywhere, they are where elk like to be. If you can learn where they like to be, you can focus your efforts on those places. Flags gave some advice along those lines. I've heard it said elk like to bed near or on benches of ridges, close to the top of the ridge.
Elk like to be at high elevation when there is little or no snow. In Colorado, the aspen/pine interface may be about 11,000 feet. Be prepared for that altitude. As a low-lander myself, I work at physical fitness, but I just accept that I'm going to be out of breath. I just stop when I need to and catch my breath. It is not a sprint but a long distance race. Be aware you need to set a pace you can maintain for 5 days or more and possibly go into over-time when you kill an elk, to pack it out timely.
It is big country, so remember how to find your way back to camp or to the truck. At night you won't have a lot of visual cues to navigate by. Having a GPS and knowing how to use it is probably a good idea. It doesn't have to be fancy or expensive.