JMH, the "zoo" I was referring to is the crowds of tourists that you would encounter in Yosemite Valley. ...sorry.
White Mountain will certainly be accessible in July. First of June is not so certain. You would only encounter a few patches of snow near the summit, and they aren't steep. I even took a bicycle to it's summit years ago.
Other fun stuff: Bodie ghost town, northeast of Mono Lake. Yosemite: Hike to Cathedral Lakes (trail starts in Tuolumne Meadows).
An easier fourteener than Whitney: Mt Langley.
Acclimation hike: near the Langley trailhead in Horseshoe Meadows: Trail Peak (sw of Horseshoe Meadows.) Or hike to (or just part way to) Cottonwood Lakes.
Driving through Death Valley is scenic, too.
You're in for a good trip!
