I'm a wildlife biologist and have worked for the F&G in Utah & MOntana and worked for other agencies in idaho and wyoming. I have hunted all over utah, montana, and western wyoming and eastern idaho and taken elk in all those states.
My two cents:
If you dont care that much about trophy bulls, then I would stay out of utah. Certain southern utah areas are tremendous limited entry hunts but the rest of the state is touch and go and you will often find a lot of hunters in the woods even in archery season. My philosophy is this...if you can hunt an area for cows or deer with a darn good chance of later drawing a bull tag in that area with great bulls, then you are making an investment in the future by hunting a cow there and learning the area that has limitless potential, vs. hunting for example a utah cow area that you may never draw a bull tag on or that unit may never produce many mature bulls.
Idaho is good in various areas but in my opinion the best areas are harder to access cold turkey without spending lots of time and scouting and tend to be in the panhandle...my idaho buddies just dont do as well on elk as my wyoming buddies.
Montana's best elk hunting is in the southwest part of the state, but if it were me, I would put in for the Wyoming non-res draw (antelope also!), then buy over the counter cow tags to get your feet wet the first year if you don't draw out on an ANY elk tag. There are lots of cow tags availabe in numerous western wyoming units (units numbering 80's through the 90's among others). Most of those units are way above management objectives. You will be able to buy the cow tag for 288 over the counter when they come available. Call the F&G biologists in that region. Plan on hiking a lot and making high elevation spike camps to find the elk. Elk in that area usually go up in elevation during september/october so prep for that sort of excersise, and dont take it lightly...build your conditioning ...its much easier to get up there and stay there for the hunt. For guys used to W/T hunting, one of the best strategies is to still-hunt dark timber in the mid-day. Elk will often filter to the top quarter of a timbered ridge - so they can bed and settle in and get wind or sight of things above them backlit against open sky at top of ridge. With good wind an experienced w/t hunter can feel at home hunting elk like this. DOnt overcall, use cow call most and occassional week bugle if your going for bull or spike. Wyoming is the place and you will undoubetbly see/hear wolves and bears oh my!. I took a buddy from Northern maine to wyoming last fall and he killed his first elk, a 5x5 the first hour of opening day, we never saw another hunter for the whole 5 day hunt there, but saw wolves, bears, cougars, and huge muleys as well as elk. The wyoming elk hunting experience is in my opinion the best true wilderness hunts, outside of maybe the bob marshall wilderness in mt.