Your input was well described, "three hikers of Level 2 experience. We are looking for easy to moderate trail hiking with packs on, 5 to 6 miles a day", but the responses received tended to be more strenuous often a lot more than a "four night, out and back or loop trip followed by a two-night out and back trip ". What you asked for is a trip no more than two easy days from a trailhead or about 10 miles.
Emigrant Wilderness is a fine area but like all areas north of Yosemite, lacks the kind of "big mountain scenery" you asked for mainly because elevations are lower than the Southern Sierra the Sierra is famous for. Also the better destinations are at least long 2-day hikes in. Geology in Emigrant is a mix of volcanic and granite but the granite there tends to be ground down to smooth forms because a giant ice cap once was right on top of it. During this droughty year water would be an issue enroute. It is however just 180 miles from the SF airport.
I'm going to give you the same advice I've given others who have never been to the Sierra and only have a short time. The destination lake is the one which graced the cover of the original Wilderness Press "Sierra North" guide book decades ago which is the most well known among mountain enthusiasts and says something about its quality as a destination. It is further than Emigrant but not that much versus many popular areas further south. And obtaining a wilderness permit via walk up or reservation is likely without having to worry about filled quotas.
It is in Hoover Wilderness in Toyabe National Forest near Bridgeport. That is 235 miles from the SF airport. The trailhead is Green Creek, on a bumpy dusty well maintained gravel road. Your destination would be East Lake at about 4 miles and 1400 feet of uphill. The second day you move up the trail to Summit Lake. There is lots of day hiking and class 2 peak scrambling in the area. It is quite colorful metamorphic geology and granite High Sierra scenery.http://mapper.acme.com/?ll=38.06637,-119.29595&z=15&t=T
Search on "east lake hoover wilderness" and you will get many more hits and descriptions. Also hits on Google Images.
As a second 2-night trip, you are going to want something closer on the drive back where you can meet the other person just arriving. On the east side of Sonora Pass is Leavitt Meadow. An easy backpack would be to Roosevelt and Lane Lakes and a bonus is the Walker River is also right there.http://mapper.acme.com/?ll=38.29243,-119.54425&z=15&t=T