Here's what I'd do (native Californian). Rent at the top and work your way down or do the trip in reverse if that suits you. Start in NorCal at Lake Tahoe. Very easy camping in an RV (Sugar Pine Point is a lovely campground). Spend a couple days as it's a nice jumping off point--easy bike rides in to Tahoe City for breakfast, the lake is very swimmable (but COLD as it's a mountain lake) with beaches walking distance from the campground. Rafting the Truckee River is nice if the water's high enough, fishing and hiking. Perhaps 2-3 days.
Then, drive down towards the valley (stop at Ikeda's on HWY 80 for lunch) and hit Napa/Sonoma County and San Francisco on your way to Yosemite. Avoid Napa on the weekend if at all possible as it's horrific traffic and crowds. There's not much to do there for young folk as it's a wine tasting mecca. That said, if you're going to splurge and eat out fancy--this would be the place. I suggest Cindy's Backstreet Kitchen in St. Helena or Ad Hoc in Yountville (where Thomas Keller often eats after work at the French Laundry).
In San Francisco there's two way to do it ($$$ and $). See if you can book the campground in the middle of the Presidio (opened last week, not sure about RVs). $$$ includes the California Academy of Sciences and a major museum (SFMOMA, DeYoung). $ includes renting bikes (Blazing Saddles) and riding along Chrissy Field over the Golden Gate Bridge then taking the Ferry back from Sausolito, walk or bike through Golden Gate Park hitting smaller/free museums (Japanese tea garden, arboretum, etc). Finish that with lunch at the Park Chalet and walk at Ocean Beach, or head to Baker Beach and look for dolphins (a rare but lovely sighting). Pier 39 is worth it for the sea lions, but is otherwise a tourist trap, so hit it early in the morning with your coffee and then high tail it to somewhere more real. Generally good theater shows. Within 30 minutes of SF dbm has the list: Point Reyes, Stinson Beach, Sausolito/Tiburon.
From there Yosemite (RV or tent cabins). Do some research and get beyond the first layer of trails, some valleys are quite deserted even in high season as they require a bit more hiking than fat Americans [shocked] will subject themselves to.
That's probably enough nature (I've not been to the Grand Canyon, but you should fit that in at some point) and you'll be screaming for some city fun. Monterey, Santa Cruz, Santa Barbara are all along the coast and merit a day's stop or break from driving if you need it. Santa Barbara has the best evening/night life, the Monterey Aquarium is gorgeous, but may feel repetitious if you hit the CalAcademy in SF).
SoCal: Hearst Castle is worth a stop. I would stay towards the coast as I-5 is boring and blergh. Cambria by the Sea is a nice town (not to pricey) as is Big Sur. Then further south you'll want to find a beach town, rent rollerblades (good 80s fun), or boogie boards. I would bet you can take surfing lessons in Santa Cruz.
In LA I'm less versed, but the Getty is lovely. Heck, go for a sporting event if you want the real feel of it.