I remember when I first moved to Utah attending a wildlife fair at Nordic Valley. During the question and answer portion I asked if I needed to be worried about mountain lions while exploring the surrounding hills on my mountain bike. He answered that I didn’t need to worry much and he doubted that I would ever see one but they were watching me all the time!
The best place I'm aware of locally to see a lion in the wild is the Oquirrh Mountain range, much of which is comprised of land owned by the huge Australian mining conglomerate Rio Tinto. It's purported to have an impressive population whose food source is the large population of mule deer and elk residing there and is where most of my limited sightings have taken place. An acquaintance of mine has the contract to transport the bi-weekly shift change of broadcast engineers by one of his snowcats from the base of the range up to the huge KSL facility on the mountain range's ridgeline during the mid-autumn to mid-spring timeframe. It's from there that most of the local TV and radio station's signals originate and has nice living facilities for the onsite staff. He and his wife, who is the snowcat operator most of the time, see lions often and the area around the road connecting the bench with the ridgeline is littered with bones of critters killed and eaten by the lions.
On a related note, several years ago a lion took up residence on the summit of Solitude's Evergreen Peak during ski season. It caused the closure of the hike-to terrain for ~15 consecutive days, during which two patrollers cautiously ascended the peak each day to verify the lion's whereabouts (usually up in a tree). I've seen two deceased lions roadside during ski/snowboard season in BCC during the past 6 -8 years: one on Torpedo Hill that had been struck by a ski bus and one just below Solitude's Entry 1 that had been hit by a Solitude employee (a woman who was a long time member of the ticket office staff).
Here's another video from southern Utah showing how to release a lion from a leg hold trap (that's not anchored to something solid, making it more dangerous) but without the assistance of a DWR officer: