A.S.
Check out the Bay area Discovery Museum in Marin. There is a baby/toddler play area that I was taking my baby to since he could walk! It is great. There are also wonderful places to hike all around the bay area which are safe for babies. You can find a book and research those hiking trails. I took my baby to the Exploratorium and let him play with stuff that he was interested in. There is also a baby gym for little kids in West Lake at the gym there. I forget the name of it but it is open on Wed and Friday mornings from 10 to noon or something like that. I always felt there was a lot to do in the bay area for kids of all ages. There is Crocker Amazon Park, which has a tiny structure for babies your baby's age. I have a 17 month old and a 4 year old and we have so many places to take them and not enough time (like another mom said). Ocean Beach and Fort Funston are another option. We can spend TONS of time at the beach, on warm days. You can even go up to Stinson Beach past Marin. Another fun thing to do is Gilroy Gardens. A perfect amusement park for your 15 month old. I took my oldest son there several times. Many of the rides are for 1 or 2 year olds. There is also the Santa Cruz boardwalk. I have a friend who takes her small kids to Great America several times in the summer. I take my kids to the Yerba Buena park, where they can ride the carousel or play on the slides and the water, which is across from the Zeum. the SF Zoo is good for a couple of hours in the afternoon. They also have a carousel, train and petting zoo inside. The aquarium at Pier 39 is fun, as well as just walking along Pier 39 and watching the sea lions at the end of the pier. There is Tilden Park in east bay, which has pony rides and a steam engine ride, as well as a FABULOUS beach at the lake. It is a great place for little kids. It is called Lake Anza. I took my first when he was your son's age. Oakland's zoo is also perfect for little kids.
We never had a shortage of things to do. I think the bay area has more stuff than a lot of other places. The California Academy of Sciences is also fun for both you and your son. There is the aquarium there, as well as a rain forest with butterflies that fly all around. He can have a lot of fun there.
I honestly feel that there are more things out there than I can get around to! I have not yet been to Lake Anza with my 17 month old son, because we are so busy doing other things, but I think I will try to squeeze it into my schedule now that I thought of it again!
There is also AcroSports, which has classes for babies around 1 to do acrobatic and floor exercises. It was a great class. There are also foreign language classes for babies, too! I took my son to a Spanish class.
Good luck and I hope you find some fun things to do with your son!