I am a non-Mormon resident physician who has lived in SLC, Utah for almost 3 years now. I work with many Mormon people (which has never been an issue) and a ton more non-Mormon people. As gung-ho pointed out, SLC is more non-Mormon than Mormon anyway. Of the 26 categorical residents in my class, I believe only 2 are even from Utah- we have people in my year from New York, California, North Carolina, Texas, Washington, Pennsylvania, Minnesota, etc.
Unless your idea of a good time is clubbing every night until 4AM (which you will probably have a difficult time doing during residency anyway), there is enough nightlife in the area to keep you busy- there are quite a few good bars, brew-pubs, live music venues, etc. There is also theatre, symphony, professional sports (NBA basketball, then minor league baseball, hockey), the Sundance Film Festival, and multiple outdoor concert series throughout the summer.
If you have any interest in the outdoors, this place is a paradise. It is all in your backyard- literally. Just behind the University of Utah hospital, there is access to mountain biking, hiking, trail running. Rock climbing is also extremely popular here. In the winter of course there is fantastic skiing, snowshoeing, etc. From the University Hospital, you can be at one of 7 major ski resorts in about a 30-40 minute drive.
As far as diversity, there are few African-American people in SLC, but there is a significant Hispanic population (I believe 8-10%) and a large Polynesian population. There is no shortage of patient population and diversity here as the U of U is a very large referral center. I routinely see patients from all over the state of Utah, Montana, Idaho, Wyoming, Nevada, and eastern Colorado.
You are probably in a similar situation that I was when I was considering residency here and people were saying 'why in the world do you want to move to Utah??!'. Of course these are the same people that have never been here and do not know anything about Utah but will say ridiculous things like- the state is all Mormons, there is no alcohol here, etc., etc. Give them a call after you have been here a couple months when you are getting a great clinical experience and also skiing in a foot of fresh powder on your day off, and you will be the one laughing...
I hope that helped clear some things up. Let me know if you have any other questions about Utah.