Balboa Park is the country's largest urban cultural park. It houses 15 major museums, performing art venues, and the San Diego Zoo. Of course there are also beautiful gardens and playgrounds. The Balboa Park website lists a changing calendar of events.
Weekends are especially fun for kids, with street entertainers and clowns around El Prado, or check out the Marie Hitchcock Puppet Theatre (performances from Wed through Friday) or the Model Railroad Museum