3:52 p.m. on April 10, 2011 (EDT)
The 'popular' hiking season in the Sierra is mid July through September. By then, most of the higher passes will usually be easily traveled and trails below snow free.
Normally on Father's Day in the Sierra there was snow fully covering trails above 11,000'. That being said, this year is a near record snow fall year.
From the western Sierra, your trails into Sequoia NP start relatively low as compared to the eastern Sierra which trails start around 9,500' to 10,500'. And a positive for you coming in from the west, is that your trails will usually take days to get to 11,000'. Sequoia is a backpacker, explores Park and has only a single road running along the western small portion of it. Unfortunately it has few accesses to the high Sierra. One is the HST.
The High Sierra Trail (HST) begins in Crescent Meadow/Giant Forrest. The first day to Bearpaw Meadows is a long day (12+ miles), but except for some undulations, generally follows a contour.
http://www.topo.com/explore?lat=36.56794345044977&lng=-118.6317138671875&level=6&type=topo
Lunch spot at Mehrten Creek (assuming an early start) and you can spend the night there or at Buck Creek or press on for another hour past Buck to Bearpaw Meadows. You will be looking across the valley at some spectacular Yosemite type domes en route. You can make Bearpaw Meadow a base camp and day hike (or move your camp) into some of the most spectacular scenery in the park on the way up to Hamilton Lake.
http://www.topo.com/explore?lat=36.55755041749668&lng=-118.56866455078125&level=6&type=topo
On a positive note, Bearpaw Meadows has a Sequoia concession run B&B of sorts. As an alternative, you can rent a horse and packers to take all of you and your stuff to a cabin with one of the best views in the Sierra from an open air bar. This means you can, if they are open this early, enjoy an adult beverage at the end of the day and kickback and relax overlooking an awesome scene...while the kids knock themselves out. If thi$ intere$t$ you, you have to make tho$e plan$ EARLY!
If you have the energy (trust me the teens will) make it up the head wall on the trail as far as you can get toward Kaweah Gap out of Hamilton Lake. You may have snow problems near the top well into late July this year. You have one creek crossing just below Hamilton Lake that will give you a bit of a concern. The rest of the crossings are bridged or routine.
You can also head up toward Elizabeth Pass for a look see down the other side. You will have snow for the top portion to the pass. You should be able to kick steps into any snow. Probably not a place for tennis shoes. Tamarack Lake should be near to snow free as well
http://www.topo.com/explore?lat=36.58230510465098&lng=-118.580810546875&level=6&type=topo
Plan on 10 days to do the HST (to the top of Mt Whitney) some late July/Aug time frame. Then would also be a good time to plan to go to Little/Big Five Lakes from below Kaweah Gap and down to Mineral King via BlackRock Pass (or several other options) then return to Bearpaw. That is a 5-6 day trip sometime.
Out of Wolverton via Panther Pass and Mehrten Meadows there is Alta Peak (area).
http://www.topo.com/explore?lat=36.58872532141109&lng=-118.6746826171875&level=6&type=topo
Or even better for some cross country is Pear and Moose Lakes and the Table Lands also out of Wolverton.
I'm making few bets that any of the above 10,500' will be mostly clear of the snow pack this early this year. But they don't call it the Flatlands for the heck of it. You should be able to walk on top of most of it - at least early in morning.
The other access into the park is from Lodge Pole and up towards Stillman Pass.
The other (Kings Canyon road end) has significant water crossing problems across Bubbs Creek. Located there is a later in year popular start on a hike to Ray Lakes and return.
If you are in the Los Angeles area the east Sierra up US395 is an easier access to the high Sierra. Out of Lone Pine might have late June possibilities as well. Cottonwood Lakes trail and the trail head in Horseshoe Meadow is a 6 mile hike to spectacular scenery at just below 11,000'.
http://www.topo.com/explore?lat=36.491139299654094&lng=-118.2127685546875&level=6&type=topo
Other forums you might be interested in asking about trails are:
http://forums.backpacker.com (SOUTHWEST Forum)
http://www.highsierratopix.com/community/ (BACKPACKING Forum)