BETA PHOTO: Map of the two climbing areas directly east of the...
This is the mountain that sits directly east of the University of Utah. The Bonneville Shoreline Trail runs along its southwestern flank. There are a couple not-very-good climbing areas on the mountain, but they're easy to access if you're at the U or in The Avenues/Downtown area.
Park at the Red Butte Canyon parking lot. To get there, turn from 700 E onto 400 S (which shortly turns into 500 S, which then turns into Foothill) and go 2.1 miles. Turn left on Wakara Way and go .6 miles. Turn left on Chipeta Way and follow it as it turns into Stover Way and then into Red Butte Canyon Road. Park at the large dirt parking lot at the end of the road, right before a gate. The Bonneville Shoreline Trail, which is used to access the climbing, is the obvious trail that leaves from the north side of the parking lot. (This is the same trailhead used to hike to the top of Mt. Van Cott.)
You could also access this part of the Bonneville Shoreline Trail from the northwest (and it would actually offer faster access to the bouldering spot), but you'd have to park somewhere on campus or in a residential area.
Both of these areas have likely been climbed at for a while. I ran across them while wandering around and they might have real names that I'm not aware of. If that's the case, let me know and I'll change them. I'm not adding any specific routes to MP but if anyone feels the need to, knock yourself out.