There are a variety of classes offered in the South Bay. Here are some suggestions:
Mommy & Me
South Bay Adult School
http://www.southbayadult.org
The Parent Education classes at South Bay Adult School are great because they are affordable, there are a range of ages and class types, and it's a great way to meet other parents, especially in the 0 to 6 month class. I did the 0 to 6 month class with both my children and also the Music, Art and Movement class. They also have classes for siblings.
Bright Beginnings & Beyond
http://southbaymama.blogspot.com/p/new-moms-dads.html
Wondernation
http://www.wondernation.com
Wondernation has a variety of classes, including my favorite - Music N Sign for babies. There are also music and movement, art, piano, guitar, preschool, mommy & me, and prenatal yoga classes. The studio is setup nicely and there is a "wonder tree" in the middle of the room that the kids love. They also do birthday parties.
Music Rhapsody
http://musicrhapsody.com/
Music Rhapsody has been around for several years and in addition to the schools around the South Bay where they hold classes, has staff that travel to schools to conduct music classes. When I was looking for a music class for my son, we tried this one out for a couple of months. He enjoyed the music and activities very much. One thing I would say is that this class is a little more structured and there are behavioral expectations in comparison to other music classes in the area, so if that isn't for you then try something a little more loosely-structured. Their website is also great for buying music or musical instruments.