I've been here a decade and my wife grew up here - I end up piggybacking on a lot of her friends and social activities, and I picked up a bunch more from coworkers and such. But for a whirlwind HN guide to the Bay Area:

Parks: check Google Maps for anything that's a splotch of green and just go to it. Local parks are free, county & state parks often charge a $5 admission (frequently not enforced). Most of them are pretty good. BAHiker.com has a good guide to state & county parks, Google Maps has pictures and reviews on local parks.

Events: Google [$city events] or check out EventBrite, Johnny Funcheap, or SFGate & the Mercury News's events sections. Also look for flyers when you visit your local downtown area, or check out the local library's website (they often sponsor a lot of fun stuff like storytime for kids or Star Wars Day). Many of the peninsula towns sponsor things like classes, swim lessons, outdoor movie nights.

Socializing: the Bay Area has a thriving Meetup scene, but tbh I've never made a friend that stuck through public Meetup groups. Had much better luck with staying friends with former coworkers, and with paid activities (gyms, courses, music lessons, kid stuff, etc.) Google is your friend here; Google [$activity near me] and you're almost certainly going to find something. Because the Bay Area is both wealthy and densely populated, it's a magnet for really good instructors: aside from the aforementioned Krav Maga gym (KravZone in Sunnyvale), there are 3 world-class gymnastics gyms (West Valley, San Mateo, Airborne); a mandolin orchestra (SF Mandolin); some hardcore cyclists (have seen them riding over the mountains in a group occasionally); and many other activities.

For mom's groups: there're some that are just organized over Facebook (oftentimes by invitation - if you get plugged into the parent networks you often find out all sorts groups), some that are public, and some that are organized through local non-profits (eg. Blossom Birth or El Camino Hospital). I'd probably start with a for-pay class to meet people (folks are often more committed in the for-pay classes, and it draws from a social class that is more tightly networked) and then ask them if they know of other resources.

Museums & amusement parks: the big science ones are the Exploratorium and Cal Academy in SF, the Tech in San Jose. Smaller science ones include Chabot Space & Science Center in Oakland, Monterey Bay Aquarium, SF Aquarium, and Seymour Marine Discovery Center in Santa Cruz. Kids ones are CuriOdyssey in San Mateo, Happy Hollow in San Jose, Junior Museum in Palo Alto, the Discovery Museum in San Jose, and Gilroy Gardens in Gilroy. Amusement parks: Great America in Santa Clara, Raging Waters in San Jose, Children's Fairyland in Oakland, Santa Cruz Beach Boardwalk, and Six Flags in Vallejo. History & general interest: Egyptian Museum in San Jose, Hiller Aviation Museum in San Carlos, Computer History Museum in Mountain View, Maritime Museum in SF, the Hornet in Alameda, and probably a few others. There are also 3 railroads that my kid loves (Billy Jones @ Vasona Park, which also has a carousel; Roaring Camp @ Henry Cowell Park; and the seasonal Train of Lights between Fremont and Sunol), and most of the local parks have splash pads that he really enjoys.