Welcome to WhitneyZone, Kurt!
The above info is excellent. I just want to add a little.
If you come in the fall, check the public transportation sites -- YARTS out of Yosemite cuts back their schedule after the summer season ends, ...about Labor Day.
The best public transportation is at Agnew Meadows and Reds Meadow (Devils Postpile area, south of Yosemite). The shuttle buses run many times a day from there out to Mammoth Lakes. Of course, then the transportation in/out of Mammoth is only once a day. If you drive a car, it is probably easiest to drop it at Mammoth, since you can get a Crest bus back to that point from any town on the Hwy 395.
Getting transportation to anyplace on the west side except Yosemite is really tough -- there is NO public transportation. And I do not know of any shuttle outfits to take you to a trail head. However, there is plenty of visitor traffic to Roads End in Kings Canyon, so hitching a ride in or out there would most likely work.
That trailhead gives you access to Bubbs Creek (west of Kearsarge Pass/Onion Valley/Independence), and also Woods Creek/Paradise Valley.
On the east side, these areas would likely be ok for hitching a ride: Whitney Portal, Onion Valley, North/South Lake.
Shuttle services never drive your car anyplace. They only provide transportation for you and your gear. As for leaving your car for weeks -- lots of people do, and we seldom hear of any vandalism.
Permits: Getting permits is a pain. If you start in Yosemite, I think you can get them easiest if you call in 6 months in advance. They go quickly. A few are held back for walk-ins and there are a few available due to no-shows.
Getting permits for any trailhead on the east side is similar to Yosemite. Except if your hike starts on the east side and ends at Whitney, you need TWO permits: One to start, and a Whitney Trailcrest Exit permit. If you start at Whitney, there is a lottery, but there are usually plenty available due to no-shows. Permits to start at Florence Lake or Edison Lake are easier to get, probably due to their remoteness, fewer people start hikes there.
