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As you may have discovered already, California heavily oriented towards the automobile, and Palo Alto is no exception. Nevertheless, there are a few public transportation alternatives that may work for you.
Caltrain provides train service to San Francisco, San Jose, Gilroy and points in between. If you are going to spend the day in San Francisco taking the train is a good alternative. This is particularly true since the introdution of the fast "Baby Bullet" trains that take you to SF in about 40 minutes, faster than driving. You can use Caltrain to go to SJC -- Mineta San Jose International Airport -- (with a connecting free shuttle bus called the VTA Flyer), and to SFO (San Francsicso International Airport)
Inside Palo Alto the Valley Transit Authority, known as the VTA, does have several bus lines, including the No. 22 bus, which runs all the time, about every twenty minutes and go along the El Camino Real corridor through several cities, all the way to Eastridge Mall in San Jose. They can be useful to get to Stanford shopping Center or University Avenue. Use this bus to get to some of the car rental places along El Camino Real just south of Arastradero Road (Hertz, Avis, Budget, etc.).
Marguerite is a free shuttle system with several lines that serve Stanford University, the Caltrain station, Stanford Shopping Center and Town & Country Village shoping Center and Stanford Medical Center.