If you would prefer to avoid both snow and skeeters, especially in a potentially higher-than-average snow year, then August or early September are actually the best times. Skeeters can last through July depending on snow conditions. Downside about August and later is that all the waterfalls in Yosemite are not as gigantic. (Yosemite Falls even dries up entirely sometimes.)
As balzaccom suggests, Hetch Hetchy or the North Rim of Yosemite Valley will be the first to become snow-free, or at least only scattered snow patches, for backpacking. Around May sometime, most years. And the waterfalls are their hugest in May. You will have skeeters though.
Grand Canyon of the Tuolumne, generally one-way from White Wolf to Tuolumne Meadows (~32 miles) is a classic and stunning trip a big later, usually June or the first two weeks of July. This year, with a larger than average snowpack I'd be hesitant to reserve a trip there before June 15.
I live in the Bay Area too. In the actual spring (Feb-April), I stick to backpacking locally such as to Pt. Reyes or Big Basin or Ventana Wilderness. Southern Utah is also terrific at this time. If you want to go to the Sierra, snowshoeing on a bright sunny day is always beautiful.