First, Utah allows sound suppressors, so you are okay on that point. Next, since a sound suppressor also acts as a flash suppressor, it will currently have to be installed on a pre-ban AR-15 rifle only. If the '94 ban lapses in September, you will then be able to install one on a "post-ban".
You will need to locate a Class 3 dealer in Utah. You can either buy a "can" that he has in stock or order the brand you want through him. I just ordered a
SRT Hurricane suppressor through my local class 3 dealer, as he did not currently stock SRT products. Since it is coming from out of state, I also have to wait on a manufacturer-to-dealer transfer, which generally takes up to 2 months.
Once the can is in your dealer's possession, you pay for it. The dealer then provides you with two Form 4's, two fingerprint cards, and a citizenship form. The Chief Law Enforcement Officer (CLEO) in your area must sign the Form 4's. The CLEO is generally the Chief of Police or Sheriff, depending on whether or not you live in city limits. You are then fingerprinted on the cards. Attach two passport photos of yourself on the form 4's and sign all the completed paperwork. Send all of this into the BATF with $200 (my dealer sends them in for me), and WAIT. It usually takes about 4 months for the BATF to approve a Form 4 transfer. When the approved paperwork arrives at your dealer's, he will notify you to come in and pick up your suppressor. Photocopy the Form 4, store the original in a safe deposit box or the like, and keep the photocopy with your suppressor wherever you take it.
Hope this simplifies things a little. Good luck.