Jeffrey S.
Yelp
You may not realize how big of a deal it is, but this is a big destination for travelers coming from all around. After an involved remodel, there's renewed interest in the museum.
To address the elephant in the room: you don't have to be Mormon to go here. You don't have to be Mormon to be interested in the place. In fact, I'd call this one of the most approachable of the Mormon destinations in downtown SLC.
Admission is free, so that's a big bonus. You will have to deal with downtown SLC parking, but I've never had much of a problem. There are just a few spots of street parking out front along West Temple, and a pay lots to the north and west. If you drive further away, you can street park for free, but you'll pay for that in footsteps; there's nothing too close.
Inside you'll find a variety of features. There's a large gift shop featuring prints and gifts based on some of the works in the museum, as well as some of the more well-known LDS works of art. The main floor focuses on the history of the LDS church, with artifacts from its past. Some of this centers around the pioneer movement where Utah was settled, and some of it pre-dates that time period to the early days of the LDS movement. There's also a small theater on this floor, so look at the schedule to see what's playing--they're just short presentation.
The second floor was quite interesting--I guess I didn't realize it was also an art museum, and much of the north and western side of this floor just features art. Some of it hold religious imagery, and some of it is just landscape, or everyday life, painted by LDS artists. I *think* this space is a rotational exhibit, so depending on when you go, you'll see different things here. The other part of the second floor is dedicated to the people of the LDS church, mainly the former presidents of the faith and holds a lot of historical artifacts from their lives. There's also a small room dedicated to the Mormon Tabernacle Choir.
I haven't been to the basement, to be honest, and it's not for lack of desire; since the remodel I've only been one time and I ran out of time. Apparently there's some movie on a larger screen there? I dunno... but it's probably worth looking into.
For Mormons and non-Mormons alike, I think there's plenty of interest here. YMMV, but it felt very much like a specialized museum and is staffed by older couples as missionaries, so I didn't see any active proselyting going on. It just seemed like a low-key, free, interesting space to go check out.