If you just want to see Monument Valley, you don't have to spend much time inside Utah itself. Just fly into Phoenix or Albuquerque, drive towards Flagstaff and then through the Navajo Reservation towards Kayenta. The park itself extends into Arizona and Utah is operated by the Navajo Tribal government. Zion, a US National Park, is also along the southern border, just a bit further west and can also be accessed through Flagstaff or Las Vegas.
As for the University of Utah, it has a lot of notable firsts including being one of the original four nodes of ARPANET along with UCLA, Stanford, and UC Santa Barbara. The School of Medicine is the only medical school in the state and has been a big pioneer in bioinformatics, genetics, and Cardiovascular medicine. It was the first Cardiovascular disease institute west of the Mississippi and was the site of the first artificial heart implant. That's about as much as I remember from my interview days and can't speak for much else except that Park City is nice (or so I hear).