Public transit seriously limits you at this airport. The only realistic way to get to Stanford is to take the free bus shuttle to the Caltrain station and then the train or bus to Palo Alto. This will take at least 90 minutes each way because the bus is slow (take the 522 "express") and the train only runs once each hour. When you get to Palo Alto, you'll have to take the free Stanford shuttle, Marguerite, to get to and from campus, and I think the express shuttle from the train station doesn't run on Saturday.
I guess then instead you take the free airport shuttle bus the other way to the trolley and go to downtown San Jose. There is an art museum and a science museum there and and assortment of restaurants, nothing special in my book.