(Fellow east-coast Bay Area transplant here)
I took a backpacking-novice friend to Lake Aloha in Desolation Wilderness last year, in late August, and he had a great time. You can camp right near the lake and awake to see the sunrise on the Crystal Range...truly stunning payoff for not a ton of mileage and fairly easy terrain.
It's not a total cakewalk - it takes some work - but it's certainly not rip-your-legs-off steep. You start the hike by the Echo Lakes, and if you want to take it easy you can even lop several miles off the trip by taking the boat shuttle (check their schedule). I think this hike is covered in Parr's Hiking the Sierra Nevada, and is certainly a popular hike. The water is sure to be pretty low due to the incredibly light snowpack we had this year. It took us about 5 hours of slow hiking from trailhead to campsite. You MUST have a permit and a bear canister, a ranger met us up there the moment we got to the lake to check both.
You won't get total solitude your first time out without putting serious trail miles down, or without going for pretty challenging hikes. Easier hikes means more people. But while I saw plenty of folks as we hiked, we camped in an area where we saw no other tents; the closest campers were about 150 yards away. You get quiet where it matters: in camp, as you eat and watch the stars appear.
I'd also suggest that you consider car-camping the first night to get used to the altitude...get up there early, get a campsite (or reserve one), and take a chilled out dayhike the day before your backpack trip. If it's your first time backpacking, or your first time backpacking above 6,000', then don't over-exert yourself the first day. Hydrate like crazy!
Another suggestion would be Kennedy Meadows to Relief Reservoir, maybe extending out to Lunch Meadow or even Brown Bear Pass to overlook Emigrant Lake (this is all in the Emigrant Wilderness). Also a great hike, lots of uphill but nothing very steep. There's a lot less people but it's a pack trail, so expect horses.
This sounds very exciting for you - have a wonderful time, no matter where you go!