Mt Olympus - Just south of Salt Lake City, great hiking trails, great views.
Timpanogos Cave, Timpanogos Cave National Monument (U.S. National Park Service)
A really cool place to visit just 45 mins south of Salt Lake City. and if you stay on utah 92 I believe after you exit the parking lot go north/northeast not back towards ogden you can follow 92 back past Sundance and you will loop around Mt. Timpanogos. Or if you are coming in from Provo traveling on U.S. 40 or U.S. 189, take State Highway 92, passing by Sundance Resort and over the mountainous scenic route known as the Alpine Scenic Loop. Because of the narrow and winding road, buses and large vehicles over thirty feet are not allowed to travel on the Alpine Scenic Loop. The Alpine Scenic Loop is closed during the winter.
This is a great spot also for hiking! Tons of aspen trees!
The Bonneville Salt Flats - just west of Salt Lake City. I have yet to visit when something major/national is going on here, but weekends there are always people running their cars/jet trucks/motorcycles flat out here and it is cool to watch.
As you can see, I have spent weeks in Salt Lake City for business so I hike evenings and weekends so I try and hit the local mountains, and yes there are tons of them. Side trips are always the most fun.
Check out Sundance if you are ever out there, you just might run into Mr. Redford in the area and if you goto Park City in late January you too can be part of the Sundance Film Festival which is really held here, and you can get some Polygamy Porter to boot!