Is their much by way of housing for families in the area? Where do most of the med students live?
I used to live on campus, from 2007-2010 to be exact. I lived in Shoreline Ridge, which is about 2 minutes on foot from the Medical School and Hospital. I lived there with my wife and two children.
Anyway, about two years ago the University of Utah HRE (housing and residential education) department decided that Shoreline Ridge would no longer be tolerating families. I say "tolerate" because that's exactly what they did for us from that point on until we all left. They made everything very uncomfortable for us, I assume to encourage us to vacate our apartments. As one family would finish up with their studies, they would be replaced with a single student. No families were allowed to apply for housing in Shoreline Ridge. At this point, there are about 140 apartments and a nice children's playground for single students to enjoy. Families are no longer welcome.
So the only option at this point for married students, or single students with children, is to live in the University Village (east and west), which is about a mile to the south, perched on the far southern edge of campus well away from the medical school and hospital. The Village consists of 40+ year old apartments which have no dishwashers, and no clothes dryer hookups. It's pretty grim compared to Shoreline.
The sad thing is that I know that there is a plan to demolish either the east or west village and not rebuild it, and then demolish the other and replace it with new housing. So there's a chance that if you do live in the Village you'll be evicted at some point to allow for either the demolition or demolition and new construction. The villages are also not off limits to single students, so you'll be competing with them to get an apartment, and if you do get one, it's one that was build over 40 years ago rather than just over a decade ago.
It's actually one of the reasons I didn't apply to the U of U. It's a great program to be sure, and I like the university overall. But it's housing department is extremely unfriendly to married students.
You can probably find a house or basement apartment to rent near campus, but the neighborhoods bordering the U of U are among the very most expensive in SLC (especially the one's near the hospital) so I can't comment on how much you might expect to pay.
Also bear in mind that the Salt Lake Valley has a great public transit system. You can ride the Trax train right to the medical school for free as a U of U student. And you can even ride the commuter train down from the cities north of SLC for free with your student ID.