Hey BB, that is one of my all time favorite movies! Thanks!
WY has some general hunt regions that you can draw every year or every other year. Check out Eastman's MRS either online or in the mule deer issue of the magazine. Just pay close attention to public land access. There are several posts on here about guys getting tags for areas rated "poor access", and disappointed that there is no public land to hunt. Also, when the area is rated "rough / extreme" in the MRS, that's for real!
I just did my first DIY mule deer hunt in WY this year and that how I went about it in general. I started planning about 12 to 14 months prior and we had a good time and I am already planning next year's hunt. The better your planning and research, the more enjoyable your trip / hunt will be.