If you have high tolerance for mosquitoes...
And assuming we have an average snow-pack this year, so that you can do this in July...
And, assuming you want to stay on trails...
Fly to Reno. Public transit down 395 to Bishop, Big Pine, Independence, or Lone Pine. http://easternsierratransitauthority.com/wb/pages/bus-routes/crest-north-lone-pine-reno.php
Hitching from the towns on 395 to any of the east side trailheads is easy and quick. If you aren't comfortable hitching, then you can hire a shuttle service.
Or fly to Las Vegas and rent a car (as Alan describes in his trip report http://www.backpackinglight.com/cgi-bin/backpackinglight/sierra-prime-trip-planning.html)
Depending on your hiking pace, hike any section of the John Muir Trail between Piute Pass and Cottonwood Pass. I.e., from the east side hike one of the east-west trails over the crest to pick up the JMT - then hike the JMT - then hike a different east-west trail east over the crest to one of the east-side trailheads.
For example, from Bishop, hitch to South Lake trailhead. Take Bishop Pass Trail to JMT. Then south on JMT as far as you have time to go. Then exit via Taboose Pass, or Sawmill Pass, or Kearsarge Pass, or Whitney Trail, or New Army Pass, or Cottonwood Pass - hitch back to 395 from the trailhead. Check with ranger station to find out which roads to east-side trailheads are open and passable - occasionally avalanches damage those roads and you wouldn't want to exit to a trailhead that has no vehicles. I wouldn't recommend exiting via Shepherd Pass, as that trailhead doesn't get enough traffic to enable you to hitch a ride in <30 minutes like the popular trailheads do.
For planning a trip in the High Sierra, nothing beats this map set:
"Forest Service 1:63,000 map set, Guide to the John Muir Wilderness and Sequoia-Kings Canyon Wilderness". Available here: http://www.fs.fed.us/r5/inyo/maps/esia/esiamaps.shtml
If you don't care for mosquitoes then you need to wait until August or September to visit the High Sierra. If we have a heavy snow-pack year and you're not prepared for snow on the high passes, then you also will need to wait until August or September.
If you are comfortable traveling off-trail, then get a copy of the map set, a copy of RJ Secor's "High Sierras Peaks Passes and Trails", and plan a trip anywhere between the Mono Recesses and Cottonwood Pass - there are no bad choices in the Southern Sierra.
Good luck, Amy