An early welcome to the Bay Area (as we call this place). I am a native Californian and adore the Bay Area. The weather is practically perfect year-round and there are a million and one free (or close to it) things to do for kids and people of all ages because you have so many fabulous micro-environments (oak forest, redwood forest, rolling hills, warm, cool, ocean, etc.).
As the previous poster said, it is really hard to help you out with so little information. Do you have any idea where you would work? If your kids are going to be school-aged then it is very important to end up in a place with good public schools. The quality various tremendously with the good schools being in more expensive areas due to our lousy way of funding schools here. Generally, the city itself is great but very expensive. Your best bet at not having to use a car is there. Avoid the northern tip of the east bay at all costs (Richmond and surrounding areas) Berkeley is nice but crowded and expensive. The rest of the east bay (San Leandro, Hayward, Fremont, etc.) is far more affordable, some areas have good schools, but it boring as heck. If you can live close to BART then you can get good public transportation to take you to the city. Unfortunately BART doesn't go all round the bay though so you are limited there.
San Jose is huge, sprawling, and has decent parts (Willow Glen, some areas near downtown, Japan town, etc.) but also has a lot of ugly sprawl, bad traffic, and areas you do not want to go. San Jose is generally more affordable. From there, things get more expensive and nicer the higher up the peninsula you go. Places like Campbell, Palo Alto, and Menlo Park are known for great schools and ridiculous housing prices. East Palo Alto is an island of affordability in an ocean of ridiculous prices but I personally wouldn't live there and definitely would not want to take kids there. Further north places like San Carlos, Belmont, and San Mateo have good schools and are slightly more moderately priced. On the Peninsula you have CalTrain running up to the city which can provide public transportation depending on where you end up working. A lot of commuters use it, but I personally don't like the train since it is loud, slow, and diesel.
Good luck and when you have more details, let us know and we can provide better information to you.