I live in Santa Clara which is pretty close to where you are. My kid is only 9 months so I'm somewhat clueless about the kid-friendly stuff but I'll give ya what I've got:
Lets see...One great place to bring kids (and a great place in general) is the beaches in Santa Cruz. Its about a 35-45 minute drive over the mountain, but very worth it. I usually go to Seacliff State Beach which is 10 bucks to get into unless you have a California State Parks yearly pass. There is also a great boardwalk in Santa Cruz, as well as a wonderful downtown area.
Free days for museums:
1st Tuesday of the month:
San Francisco Museum of Modern Art
The De Young Museum (SF)
The Legion of Honor (SF)
1st Wednesday is the Exploritorium (SF)
1st Sunday is the Asian Art Museum (SF)
3rd Wednesday is the Academy of Sciences (SF)
Plus if you have a Bank of America card they have museum days at the San Jose Tech museum and the San Jose Museum of Art on the first weekend of the month: http://museums.bankofamerica.com/
So far the best farmer's market I have been to (outside of the SF one at the Ferry Building...which is a bit overwhelming really) is the one in Mountain View held on Sundays from 9-1 at the train station by Castro Street.
There is a Whole Foods near to you in Campbell on Bascom and Hamilton, but there is a nicer one in Cupertino on Stevens Creek Blvd near De Anza. Trader Joe's rocks and is cheaper than Foods Whole (but doesn't have as good a selection of bulk items sadly).
There is a great baby store called Tiny Tots baby boutique in downtown Campbell: http://www.tinytots.com/
They are also a diaper service. Downtown Campbell also has a farmers market I believe. They also have a place called Harmony Birth Resources that offers classes and such for babies toddlers and kids: http://www.harmonybirth.com/newfamilies.html
There is a pretty good kid's second-hand store in Santa Clara called Kid-2-Kid. It is on Homestead Road by Kiely. There is an outlet mall in Milpitas called The Great Mall (how creative), and another further one in Gilroy. If you are ever down by Gilroy (say on the way to Monterrey) its worth stopping just for the American Apparel Outlet. They have kid's clothes there pretty cheap.
Crap, I just wrote you a novel. Anyways welcome!! I can't tell you much about neighborhoods because I have only lived down here for about 7 months.
Oh, one last thing is that Las Madres
is a local mothers club in the area; there is also a babywearing club and a local chapter of the Holistic Mom's Network
(which I haven't checked out yet). PM me if you want more info about these.