I'd look to the Avenues, Downtown, Foothill or Sugarhouse areas are the best places to live in terms of cultural diversity and public schools. There are also nice places to live in Holladay, Milcreek and Sandy, though their public schools aren't held in quite the same regard and the culture is more homogeneous (though this is changing every year as people move around the area).
If you're looking for private, non-LDS schools, there are some good ones available, depending on the age of your kids.
There are two heavy-hitters in the private school realm in SLC:
- Rowland Hall is a great place with excellent teachers and a long-running history of academic excellence, offering pre-K through 12 on two campus locations. They also have an integrated ski racing program, Rowmark, that includes a full-on ski racing academy for grades 9-12. Rowland Hall is located near the University of Utah, between the Avenues and Foothill neighborhoods. (Full disclosure: I'm an alumnus of Rowland Hall and Rowmark Ski Academy.)
- Waterford School is located in Sandy (convenient to both BCC and LCC) and has a great academic reputation. They don't offer an integrated skiing program like Rowmark, but students who want to pursue ski racing from Waterford often find spots on teams from Snowbird or Park City.
If you have high school age kids, Judge Memorial High School is a Catholic school that has a great reputation and is located downtown (a few blocks from Rowland Hall/Rowmark).
Another school option in SLC is the Open Classroom, a charter school for K-8. I have many friends who are alumni of this program, and it has a very high success rate and a great academic reputation.
Hope this helps!