There are many overlooks in which to park a car for spectacular views of the city. Streets like Chandler Drive, and Wasatch Blvd are fantastic.
For a close view of the city and great place to watch the sunset over the Great Salt Lake, Ensign Peak is minutes from downtown. Other close hikes are Avenues Twin Peaks and The Living Room, which has several rock slabs configured in the form of chairs and couches for relaxing while watching the sun drop behind the city.
For peak baggers, the skyline goes from smaller Black Mountain and Big Beacon to Grandeur Peak and Mount Olympus, to the towering Broads Fork Twin Peak and Lone Peak where you will feel like you're on top of the world.
Check out the view of the city, surrounding mountains, and the Great Salt Lake at sunset. This photo was taken from above Capital Hill. We drove past the right side of the capital and continued up. Eventually we made a left turn (it wasn't too far...) and found a small park within a residential neighborhood. The was a short walk up to a viewing platform.
Sun Rise. Best sunrise I have ever seen facing east was one morning in August. We were coming accross the flats from Reno the night before and got to see the sun come up through the pass on the east side of town. Definately recommend Salt Lake in the morning.
There's still snow on the peaks at the end of May.
This mountain range provides a great backdrop to the city.