Staying right on the University of Utah campus, and walking around during the day, gave me a nice feel of a heretofore unknown-to-me campus. The University of Utah is nestled right along the foothills, and has fantastic trails just blocks away from the dorms.
Within just a couple miles, the intrepid undergraduate can gain a 2000' and summit Mt. Wire. It's also popular to head to the top of Red Butte Peak, an unofficial sub-summit with a more direct view of campus. I ended up there, by hitting the saddle with the Mt. Wire summit, and then enjoying a technical, rocky ridge run back along the spine toward Red Butte.
I only explored for a few breakfast miles, and wound along one of the small, shaded canyons of the Bonneville Shoreline Trail, but even more miles and trails abound from campus, from very runnable crushed gravel to steeper, technical trails. It seems like just about the most ideal trail-running setup near a major University campus in the lower 48 that I've seen, or at least up there with Boulder (which gets a slight nod above Ft. Collins for accessibility of summits)...what other contenders are there for "best trails near a major public university?"
It's too bad the bike infrastructure could/should be way better in SLC.