3 Weeks Excursions in State Of California

Hi Tejas! There are a LOT of places I would recommend visiting since you are so central in the bay area. I would recommend:
San Francisco
1.  California Academy of Sciences for museum. All of  Golden Gate Park is wonderful and you can spend a whole day roaming around and visiting the different museums. There is also the  Japanese Tea Gardenand   De Young Museum very close to the museum.
2.  Exploratorium. It is located on the Embarcadero. Brand new (within the past year) and has a lot of hands-on activities to do for both adults and kids. 
3.  Ferry Plaza Farmers Market on a Saturday morning
Outside San Francisco
1.  Muir Woods - See the HUGE redwood trees.
2. Take the Ferry to Sausalito. Best to go when its a clear day so you can see all of the bay, SF, Golden Gate Bridge etc.
3.  Santa Cruz - visit the beach and has a boardwalk with rides, roller coasters and arcade games. Or a more low key beach area to visit is  Capitola which is pretty close by. 
4.  Walnut Creek has the most nightlife in the East Bay near Lafayette. You can also try some places in  Oakland. 
I am sure i will think of some others and update when I do!
Nicole

For your beach fix, I'd highly recommend taking a day trip to Santa Cruz Beach. They have a great Santa Cruz Beach Boardwalk that's not only fun, but also filled with history- they are the home of the famed 85-year-old red wooden rollercoaster, 'The Giant Dipper'. While there, you MUST stop at Marianne's Ice Cream‎. This is an old classic sundae spot that has been around for over 50 years- the line is alllllways out the door, even in the middle of winter (they have everything from basic vanilla to coffee cardamom crunch flavors, which is my personal favorite). In San Francisco, the Ferry Building holds a Saturday morning market that is unsurpassable. The hustle and bustle is vibrant with people passionate about their city, food, and community. Nearby the Ferry building is the Exploratorium- it's a type of 'strange science' museum that offers inside looks and hands-on demonstrations of the most imaginative, quirky things. It's a very kinesthetic experience that would turn even the most science-despising person into an aspiring scientist for the day. Mount Diablo State Park in Danville (California) (which is close to Lafayette) is a great place for a hike on a Sunday morning. You can see panoramic views of the entire bay area, observe the rolling green topography, and have a picnic as the sun goes down. There's an amateur rock-climbing wall there, too, that is actually doable, even for beginners. For a jungle safari you have to check out Safari West. It's hidden in the heart of the beautiful wine country, and houses hundreds of species of animals shipped from Africa, free to roam about the unenclosed acres upon acres of land. Your family gets to take a safari through this area, getting up close to water buffallo, giraffes, cheetas, rhinos, and the like (warning: if you are afraid of animals, it probably isn't suitable as this is a very up-close-and-personal experience). I hope this helps! -Carina

Folks already suggested some great local attractions. If you are up for a bit of a drive:
1) Yosemite National Park is amazing that time of year. It will be crowded, so plan ahead, but it is a truly amazing park.
2) Napa Valley is not too far away. There are wineries of course, but a couple great places for kids too are Oxbow Public Market and  di Rosa Preserve. 
3) The Monterey Bay Aquarium is great fun and you can continue your drive south along highway 1 through Big Sur. The drive and scenery are breathtaking. Being a vegetarian in the bay area is very easy. Almost every decent restaurant you will go to will have options. And if you are unsure, simply call ahead. Have a great trip!

Muir Woods is a wonderful. The Exploratorium -- MUST GO. I still have memories of going when I was a kid (well over 30 years ago). It's that good -- really (call ahead and check their schedules to see if any special events are happening). Highway 1 is a beautiful drive -- explore along the way. There's whale watching in Monterey Bay depending on migration patterns -- or there's always the Monterey Bay Aquarium. Definitely worth it. There'sCalifornia's Great America for an amusement park, Raging Waters not too far away, and also the unique adventure of  Alcatraz.  Or the calmer and more literary Jack London State Historic Park.
If it lines up with Gilroy's Gilroy Garlic Festival Association make the trip -- you'll smell it from the highway but it's one of a kind. For Vegetarian food, you've got some great options. My favorites in the Bay Area are Herbivore (absolutely LOVE LOVE LOVE this place -- it's Vegan food but most of the time you won't be able to tell), Greens Restaurant is a bit more expensive but is a nice night out (maybe without the kids?), and Cafe Gratitude -- if you want raw food. Saturn Cafe and Dharma's have you covered in Santa Cruz for relaxed environment and vegan/vegetarian food. Per beaches - there are as many kinds of beaches as there are personalities. My favorite is Natural Bridges State Beach near Santa Cruz -- you can watch the surfers around the bend at the other parks but this has tide pools you can wander at low tide. If you catch it in the right season, you'll also get to see the Monarch Butterflies as they migrate. Further down the coast, you'll find Ano Nuevo State Reserve where you can see sea lions galore. Just remember the golden rule of summers in California -- if it's hot inland, it's cooler on the coast but it'll be cold in San Francisco. If it's too cold in SF, go inland.

As Nicole has said, there are a LOT of places. Here are few additions

• LA metropolitan area - Universal Studios Hollywood, Six Flags Magic Mountain and Hollywood - about 4 days (could be much more)
• Las Vegas and Grand Canyon National Park (3 days)
• couple of days in San Francisco city (2 days)
•          Ripley's Believe It or Not! Museum
•         Coit Tower
•         San Francisco Chinatown
•         Ghirardelli Square Shopping Center
•         Pier 39
•          Alcatraz
•          Golden Gate Park
•         Lombard Street
• There are no end to the list of places for shopping - Factory outlets in Gilroy Premium Outlets and tracy, all superstores
• If you have time, visit any of the Trader Joes stores. It is an interesting place (at least I love it)
• Some standard places like The Mystery Spot, 17-Mile Drive (south of San Jose)
• In California, there is no dearth of Indian restaurants where you get good vegetarian food.