To clear up some misinformation, Mt. Whitney is quite an extreme and unique example of somewhere one might not be able to go backpacking on any given day in California, but still if someone who applied to reserve a permit earlier this year failed to show up on their appointed day you or I could still walk into the Lone Pine Ranger Station and pick up that permit on a first come, first served basis. http://fs.usda.gov/wps/portal/fsinternet/!ut/p/c5/04_SB8K8xLLM9MSSzPy8xBz9CP0os3gjAwhwtDDw9_AI8zPwhQoY6IeDdGCqCPOBqwDLG-AAjgb6fh75uan6BdnZaY6OiooA1tkqlQ!!/dl3/d3/L2dJQSEvUUt3QS9ZQnZ3LzZfMjAwMDAwMDBBODBPSEhWTjJNMDAwMDAwMDA!/?navtype=BROWSEBYSUBJECT&cid=stelprdb5150055&navid=160100000000000&pnavid=160000000000000&ss=110504&position=Not%20Yet%20Determined.Html&ttype=detail&pname=Inyo%20National%20Forest-%20Recreation%20Passes
Furthermore, all trails in Yosemite National Park are on a quota system whereby 60% of the permits can be reserved ahead of time while 40% are on a first come, first serve basis (it used to be 50/50) so there's no reason to cross that area off your list of choices: http://www.nps.gov/yose/planyourvisit/wildpermits.htm
Since you neglected to mention where you live and how far you're willing to travel and California is a wonderland of backpacking opportunities in every direction I can only be quite general in my recommendations.
And although you neglected to mention your backpacking experience I take it from your complete lack of knowledge of any backpacking trails or destinations, your very general goals, and your lack of initiative in researching this much needed information other than asking this question, that it would almost be irresponsible for someone to direct you to head out anywhere into the wilderness this obviously unskilled, un-equipped, and ill-prepared.
July and August is the height of summer and the prime time to go backpacking in the Sierra and other mountain ranges in California so most people who backpack have planned their vacations for this time, so that's something. You'll definitely want to stay away from hiking in any deserts this time of year. So mountains it is!
The mountains get cold at night and it's been known to rain in the mountains in the summer so you and all your friends will have to have synthetic sleeping bags rated down to 20 or 30 degrees, the lighter the better. It's unlikely you can afford down sleeping bags which do not keep you warm when wet like synthetic, though you'll want to keep your synthetic bags nice and dry as well to be on the safe side. A large black plastic garbage bag liner inside your backpack should suffice. You should bring another large black plastic garbage bag to use as a poncho should it start pouring rain.
There are microbes and protozoa and other things in the water in the mountains so you'll have to have a water filter or water purification chemicals of some sort. Boiling water is not going to cut it because you'll be doing all your cooking over a backpacking camp stove and your fuel must be saved to heat food, not wasted on boiling water. Much of the High Sierra has restrictions on campfires so the higher you go and the more popular area you go expect to have a cold camp, camp stove only.
You'll need a topographical trail map of the area you'll be visiting and compass and to know how to use them. http://egsc.usgs.gov/isb/pubs/factsheets/fs03501.html
Buying, preparing, repackaging and packing food for six people for three nights/four days will be an expensive and daunting experience, yet a good grocery store has all you'll need. Here's some nutrition information: http://www.thru-hiker.com/articles/trail_foods.php
Black bears are also a problem throughout California and you should and in some places must protect your food from bears either by hanging or storing all your food in bear-proof canisters which can usually be rented at the National Parks that require them or from your local outdoor store such as REI. I absolutely despise the trouble of hanging food (finding a tree with a branch at the right height, finding a rock, attaching a rock to a line, throwing a line with a rock over a branch, counter-balancing everything, bah!) and so I carry my $250 Bearikade bear canister everywhere I go, which holds enough food for two people for a few days.
Places to go in California: The Sierras, from King's Canyon/Sequoia National Park and surrounding Wilderness Areas such as the Golden Trout Wilderness and others, to Yosemite National Park and surrounding Wilderness Areas including Emigrant Wilderness, Hoover Wilderness, John Muir Wilderness, Ansel Adams Wilderness, the Lake Tahoe area including Desolation Wilderness, Mokelumne Wilderness, Carson-Iceberg Wilderness, and farther north than I've been all the way to Lassen National Park, Shasta, Trinity Alps, and the back down the coast to Redwoods National Park and other state parks, to Point Reyes National Seashore, and down the coast where that other guy's been I guess. It won't be as cold along the coast as in the Sierra of course but still it's often foggy in northern California.
Specific places to go: Loch Leven Lakes trail off HWY 80 east of Sacramento in El Dorado National Forest, no permit required except a Campfire Permit. Trail to Gilmore Lake and Mt. Tallac in Desolation Wilderness adjacent to Lake Tahoe, $5/person permit required. Red Bull Lake Trail into Carson-Iceberg Wilderness off HWY 4 east of Lake Alpine, free permit required. From Glacier Pass Road in Yosemite NP the trail to Royal Arch Lake, free permit required, $20 entrance fee.
I have Sierra North and Sierra South with 200 trails between them plus trail guides to Emigrant Wilderness, Carson-Iceberg Wilderness, Yosemite NP, Trinity Alps, 100 Day Hikes in California, the High Sierra Route, Secor's The High Sierra, topo maps, and dozens of links to trail descriptions online. I mean where do you start? I suggest you go to the books store or library and start reading.