Well pwacannon, you found the right place to find out a wide variety of information. Most of the folks here are more than willing to help a fellow hunter get their feet wet for the first time, or after a long dry spell.
Hunting Mulies and/or elk is a different ball game than hunting whitetail deer, but it's a passion you will fall in love with if you are serious about getting back to the hunt. First thing you should do is make sure your hunter saftey card is valid here in Utah, or if you don't have one, take the course and get one. In fact, I would suggest you take the course anyway to brush up on a few things, it will surely help. Go here to learn more: http://wildlife.utah.gov/hunter-education.html
There isn't any need to find private clubs or even to hunt on private land here, there is more public land than you could ever hunt in your lifetime. The application period to draw a deer tag is over for the year, but there is still a chance to get a left over tag later, if you chose to do so. Personally, if I were you I would spend this year bringing yourself up to speed on hunting mulies and exploring the various hunting units. Spend lots and lots of time on the DWR website, there is a ton of great info there: http://wildlife.utah.gov/
I would also spend a lot of time on this site searching the archives for information. There is a search function that works fairly well. If you chose to become more than a one post wonder, don't be shy about posting up questions here, many good folks will help out. There are also some really great books written about hunting mule deer and elk. They are also here somewhere in the archives :mrgreen:.
Whether you want to hunt with bow, muzzie or rifle, I'm sure you will find some hunts that will tickle yer fancy. And don't forget about upland and small game. They are, for the most part, plentiful and fun as all get out to chase. You may even want to go after a coyote or two to get your blood going again, and make fifty bucks per to boot. That's on the dwr website as well.
If you are the bashful type, feel free to send me a pm and I will do what I can to get you headed in the right direction. Best of luck to you and welcome to the forum...