Redwood Park in Oakland is a great place to go. It has picnic tables and a small play structure. And, you can take a walk among some very beautiful redwoods - easy, flat and paved so good if you need to use a stroller.
I'm a huge fan of Crissy Fields. There's a ranger station where you can get information, a small cafe if you decide you don't want to cook, and tons of places either on the grass or on the beach to sit and share a picnic. There's also public bathrooms, and a nice walkway to Fort Point. There are several parking lots that can get full, but even so, parking isn't generally too horrible. My kids LOVE going there, since there's beautiful views of Golden Gate Bridge and they can run and put their feet in the water. There's also plenty of grassy space for games.