Give Bell's Canyon a try. It is a great place to visit, and is very close to the Salt Lake Valley. It is a great little hike on as well.
There is a good description of the climb here:
Here are the basics:
The rock on the Bells is generally good near the bottom, but deteriorates higher up on the walls. Be prepared for more of an alpine experience (except with chain anchors!) than in the Cottonwoods. Crowds seem to be kept away from this canyon because of a long approach - at least an hour for any climb. The Bells face south, so the walls get hot in the summer. However, the altitude is higher, so it takes longer for snow to melt in the spring (probably melted out by April).
Exit on Wasatch Blvd. from I-215. Follow Wasatch past Big Cottonwood Canyon heading south. Stay on Wasatch Blvd. as it approaches Little Cottonwood Canyon (requires a right turn at a stoplight - don't go to LCC). Look for a parking area on the east side of Wasatch at 10245 South. This is the trailhead, which has changed since the printing of Ruckman's book. From here, a popular hiking trail climbs steeply east into Bell Canyon.
This climb is quite near my home, and is a local favorite. It is well worth the short hike.