Taboose Pass may not be your first choice of places to start your hike but it will certainly get you to some remote places. Many people chose this pass to do an easy (if you can call it that) hike to the top of Mt Sill(14,062ft) Its rated at a class 3 so no ropes are needed. Its a lot of uphill as you start out doing about a 6000 ft climb just to reach the top of Taboose Pass and most of its loose talus with few places to camp but I enjoyed my two nights nearby. You could use one of several individuals listed to drop you off here.
To the East Side of the Sierras - There has been a big change in the Inyo bus service for the better. There is now a bus that goes between Lancaster and Mammoth. The bus leaves Mammoth at 8:05 AM and arrives at the Lancaster Greyhound/Metrolink Station at 1:15. At 2:30 PM it starts its return trip to Mammoth. It runs Monday Wednesday and Friday. Another bus runs between Lone Pine and Reno, stopping at the Airport and Greyhound Bus Station. The Bus Station is a short walk from Amtrak. It departs Lone Pine at 6:15 AM and arrives at the last stop at 12:15. It then departs Reno at 1:20 for its return trip. It runs Monday, Tuesday, Thursday and Friday. Neither of these buses run on major holidays. There are also a number of buses running between Bishop & Mammoth, Lone Pine and Bishop, Bridgeport and Carson City etc. These buses run several times a day so check their website to see if something works for you. The Part of their route also overlaps with YARTS in case you wanted to get to Yosemite Valley. Fares were not available when I last looked.
To trailheads on the eastern side - If you're heading to Mt Whitney you might be able to hitch hike from Lone Pine. I've hitch hiked in Yosemite with mixed results however, I've had excellent results hitch hiking along Highway 395. I had a backpack and had just finished hiking 175 miles so I looked the part however, two guys in Bishop that weren't backpackers and were just hitch hiking. They told me no one was picking them up. I walked to the edge of Bishop and stuck my thumb out and had a ride in 15 minutes from a couple getting ready to go hiking. I was hitch hiking from south of Big Pine to Yosemite and my first ride took 30 minutes and the rest took about 15 minutes. If you don't want to hitch hike you can try this list of shuttles provided by the Sierra Clubs Loma Prieta Chapter. There are several people that run shuttles to trailheads but don't expect them to be cheap.
YARTS - Yosemite Area Regional Transportation System, they make runs from Merced to Mammoth and stop at several trailheads.
INYO MONO TRANSIT - Routes from the Reno Airport down highway 395 to Lancaster Greyhoud/Metrolink station.
RENO/TAHOE INTERNATIONAL AIRPORT - the airports website
www.395.com - anything and everything having to do with the east side of the Sierras and highway 395.
TRAILHEAD SHUTTLES - A Sierra Club site that list shuttle information for the highway 395 and to the trailheads.