Lots of good suggestions above, but to optimize your experience it would help to know how much of the Sierra you have seen in your past trips. Mokelumne and Carson-Iceberg are nice for avoiding crowds (I’ll be going down there soon), but IMO are not as unique and different from where you live as other options. The open granite of Muir’s “Range of Light” is mostly in the high country, from Sonora Pass or the Sawteeth down to Whitney, but Desolation is a nice miniature version. As noted above it can be crowded, being one of the most visited Wilderness Areas, but you can avoid crowds by going in from the south. From Reno you can get to the Twin Bridges trailhead in about 1.5 hrs, with nice views of Lake Tahoe on the way. Pick up a permit in south shore, which would likely be available because most people go in from the easier eastern trailheads. A 2000 foot climb (partly cross country) takes you into Desolation Valley proper with lots of open granite and areas to avoid crowds. Tomharrisonmaps.com sells a great map. One advantage of this trip is that it only goes up to ~8000’, so acclimatization would not be a big issue, especially if you have already been in Reno at 4500’. As a disclosure, BTW, while Desolation is close I don’t go there much because of the crowds. Others here know it better. Any comments from Rick D? Also, thanks to Scott for the Showers Lake details; sounds interesting. Note, though, that it isn’t in Desolation, but SE between Echo Summit and Carson Pass.
While Desolation is close to Reno, if you haven’t been in the High Sierra you should consider driving a few hours down the east side for a trip into the high country. With three days you could go into the Ritter/Banner area, the Tuolumne area of Yosemite, Rock Creek, or perhaps the Evolution area out of Bishop. Trailhead quotas can be an issue, so one approach would be to drive down to the Mono Lake visitor center where you can get recommendations, a permit and maps for many different places. You can also get permits by mail.
Note that our snowpack this year is twice normal, so there will be lots of snow (and bugs) in July at higher elevations. While this may limit access some places, it also opens up campsite possibilities with snow for water. Hope you have a good trip.