Hi, AC Transit provides some crossbay service.
Line "DB" (Dumbarton) goes from the East Bay, Union City BART, to Palo Alto Caltrain and Stanford.
Line "U" (University) goes from the East Bay, Fremont BART, to Stanford
Line "M" goes from the East Bay, Hayward BART, to Hillsdale Caltrain station
Another alternative is to take BART to Millbrae and transfer to Caltrain there.
If you are coming from UC Berkeley, this is the fastest and most flexiable (IMO) way, though it is the most expensive.
Upgrade cost from local fare to Transbay is 1x local fare.
http://actransit.org/riderinfo/busfares.wu
You can also try the
www.511.org trip planner. Its fairly straightforward to use.