I should let the real UT guys answer these questions, but I'm bored so I'll put in my .02...
Any place that serves liquor and not much food (ie a bar) is a private club, and you have to have a membership. You can get drinks at restaurants, but as I understand it, you have to completely finish one before receiving another.
Also, you can only smoke in private clubs, which is VERY nice. No smoking in restaurants, bowling alleys, etc.
I was underage at the time, so I can't really comment on the club scene.
SLC is in a huge valley and is surrounded by mountains. The Wasatch range, which is home to all the big ski areas, is 1/2 hour or lesss from anywhere in the valley. World famous Alta ski area is literally eight miles from the suburb of Sandy. How's that for easy access? The Great Salt Lake is at ~4400' and the summit of Snowbird is 11,000'. The mountains rise very abruptly out of the valley.
As for cool neighborhoods...I would guess something near the U. of Utah would be best for a young single person. The U. is against the mountains near I-80 with easy access to Park City, which allegedly has the best nightlife. I lived in Midvale and everything I needed was close by. Cheap rent, shopping, dining, entertainment, etc. I could be at Alta in 1/2 hour from my apartment at 72nd & State. It was beautiful. In Denver, it takes a minimum of an hour to get to decent skiing depending on where in the city you live.