I had actually gone through the area we now hunt, several years ago when my wife and I were on vacation. She and I both like to go out West, many times have I taken back road excursions and she never fusses. Sometime we would pull over in a good looking place and get the binoculars or a spotting scope to see what we could spot. I also try to keep up with where good animals come from, I knew the region we currently hunt had potential for good muleys and whitetail. After exploring the area personally I knew I wanted to hunt the area if I could draw the tag. When you find a good area, the best person to call is the game warden for that area. The wardens I have contacted with WY G&F have always been more than helpful. If you explain to them what your looking for,(ie. big mule deer)they can give good insight into your chances of finding what your looking for, where to look public vs. private and landowners in that area that allow hunting. I would dare say that wardens from any state that you contact will be just as helpful. I have also received excellent help from G&F personnel in NM. The state biologist for the area can also be of great assistance. You can also attain landowner list from the chamber of commerce that is closest to the area you want to hunt.
As far as the drawing process is concerned, most of the states out West are on the draw. Some states still have some tags that can be purchased over the counter. Also most of the states have some type of point system, therefore when a hunter puts in for a particular tag and fails to draw he/she will earn a point. The more points you have going into the draw the better the odds of drawing. The prime tags are hard to come by for some species and if you ever hope to draw such tags you had better be earning points. I actually put in for several states each year, hoping to one day draw Bighorn sheep, Elk, Mule Deer and Antelope tags in the limited areas. There are states that have areas in which a hunter can draw a tag each year or at least one tag every 3 years. I will admit it takes a LOT of homework as far as researching hunting areas. The only other states that I have hunted so far is NM once for deer and once for elk and AK to hunt dall sheep.
The trespass fee that I mentioned is confusing to us southern hunters. It is simply what the landowner charges the hunter to hunt on his/her land. The particular piece of private land that we hunt cost $250/person and we generally can stay as long as we want. I have seen where a landowner might for instance charge $300 for 3 days of hunting, it can vary from landowner to landowner and in some instances greatly. Most of these landowners are farmers/ranchers just trying to supplement their income. I also go and knock on doors to inquire about a good looking piece of property.
I see that you are still a young man, if you really want to someday hunt certain game animals out West in the prime areas-- start applying. I wish someone had told me the same years ago. I am now 45 and I started putting in for a couple places when I was 35, now I'm building points in AZ, NV, UT, MT and WY. Hopefully someday I will hit the lottery with a good tag for some of the species that I am applying to hunt. Hope this answers your questions.