3:27 p.m. on June 12, 2010 (EDT)
I'll second Sierra (SEquoia, KIngs Canyon NP --SEKI). Mineral King to the south of SEKI should not be overlooked as well - it is a relative newcomer to Sierra's Three Graces (Yosemite, Sequoia, Kings).
SEKI has fewer hiking visitors (but still quite a few) than Yosemite and darn few roads.
On the WEST side, Sequoia has a road skirting along the western border. There are only a few trail heads leading deeper into the park. One, Crescent Meadows leading up to Hamilton Lakes are (spectacular!)
http://www.topo.com/explore?lat=36.56397269253418&lng=-118.583984375&level=6&type=topo
is the start of the High Sierra Trail (5-9 days one way) crosses the sierra and ends at Mt Whitney. Kings Canyon has a single road leading in dead ending at Cedar Grove. From here there are beautiful hikes east through Rae Lakes and back (VERY POPULAR unfortunately), North that goes up a tortuous first day to the Ropers Route north mostly cross country to Yosemite; or south over Avalanche Pass and into more of the park and the first picture of Iambertiana's.
Out of Mineral King you can go over Black Rock Pass, after visiting Five Lakes area:
http://sierrahiker.home.comcast.net/~sierrahiker/FiveLakes/5L13.html
The map of the area is:
http://www.topo.com/explore?lat=36.472785266221244&lng=-118.5489501953125&level=6&type=topo
On the EAST side there are no road accesses to the parks, but almost every wide spot on US 395 has a trail head access to a pass that gets you directly into SEKI on the 'back bone' of the high Sierra.
There are many long trip options to choose from on the east side. Most connect up with either John Muir or Pacific Crest Trails. All trail heads have quotas to regulate how many visitors you will encounter on the trails. The entry paths are all well worn by day hikers, but once over the pass you can plan on routes that will have very few other hikers on it. The JMT will always have traffic. Mainly because it goes through some of the very nicest places on earth.
One east side project (5-6 days) puts you in at Cottonwood Lakes (trail head starts at 10,400 from the parking lot) up to the lakes,
http://www.topo.com/explore?lat=36.4907958148913&lng=-118.220947265625&level=6&type=topo
then over Army Pass down Rock Creek and up to behind Mt. Whitney on the 3rd night
http://www.topo.com/explore?lat=36.556864035917606&lng=-118.3497314453125&level=6&type=topo
at Crab Tree Meadows which could be a jumping off spot for a day hike of Mt Whitney from the 'back side'. (Timberline Lake - one shown in Iambertiana's pictures above - is just east off the map)
Continuing north you cross Wallace Creek and look west at the Kaweah (op cit. Iambertiana) and once across Big Horn Flats (awesome spots for Kodak moments) you get into the upper Kern River drainage area. One of the best all around views in the Sierra from near Tyndall Creek your next stop.
http://www.topo.com/explore?lat=36.64609147173674&lng=-118.38629150390625&level=6&type=topo
From here you can day hike around the area up to some of the passes on the west or east up Tyndall Creek to Shepherds Pass (and even Mt Tyndall) for a grand view of the entire area. Either of these hikes are a nice day hike.
The next day its a spectacular day you pay for by going over Forester Pass,
http://www.topo.com/explore?lat=36.70277877705462&lng=-118.36749267578125&level=6&type=topo
but uncompromising scenery going down the north side to Vidette Meadows your last camp.
You exit east out and over Kearsarge Pass to Onion Valley (the pictures of Kearsarge Lakes are in Iambertiana's pictures as well.
http://www.topo.com/explore?lat=36.772528021353864&lng=-118.37841796875&level=6&type=topo
From here you have to figure a way back to Lone Pine and up to Horseshoe to retrieve you car.
BUT, if you go counter clockwise from Onion Valley over Kearsarge and Forester Passes to Tyndall Creek (and sight seeing), then up over Shepherds Pass down to the trail head at Symmes Creek it is a dusty three mile hike to the road that leads up to Onion Valley and a simple hitch to your car for a loop. This would be considered a fairly strenuous hike because of the altitude you will be at and the long trip down from Shepherds Pass to Symmes Creek.
Other trips on the east side can be just overnights or two to some extremely nice Sierra high lakes. Such as Big Pine Lakes west of Big Pine
http://www.topo.com/explore?lat=37.12548094234295&lng=-118.49078369140625&level=6&type=topo
then out and up to Bishop for an few days from South Lake Trail head south over Bishop to Dusy Basin and maybe a day trip over Knapsack Pass.
http://www.topo.com/explore?lat=37.104357697668036&lng=-118.5494384765625&level=6&type=topo
to just west of the BIG mountains you were looking at from the east at Big Pine Lakes
or, or, or.......
or, or.....