Hey there. My wife and I live in SLC, I've spend most of my life here.
Since we are a lesbian couple, our top priority in choosing a neighborhood was one that was had other LGBT folks and was more liberal minded. So our priorities might not be what is important to you.
The LDS church is very big here (as I'm sure you know) and can be very overwhelming and a culture shock for a lot of people. I grew up in suburbia and many kids couldn't play with me because my family was not LDS. I know this is an issue still for many people and their kids, but definitely not all.
I'm a big fan of Sugarhouse, the Liberty Park area and The Avenues, they are all on the more expensive side though. The Liberty Park area is the most affordable and is more diverse.
I'd stay away from South Jordan, Draper, Herriman and West Jordan if it were me. How old are your kids? The public school system here is not the greatest, but there are some better schools.