Currently there are two train services to Downtown San Francisco - BART (LINK ERROR) and CalTrain (LINK ERROR). Neither depart from the airport itself but they both provide shuttle transfer to the stations.
The BART system (LINK ERROR) departs Colma BART station, the nearest CalTrain (LINK ERROR) station is Millbrae.
There are many operators of bus/van services from SFO.
The SFO Airporter has departures every 15 minutes, Sam Trans bus every half hour. Departures are from terminal Arrivals areas. Cost to downtown San Francisco: Sam Trans bus $2-3, SFO Airporter $10. http://www.sfoairporter.com
Door-to-door van services depart from each terminal. As you clear the baggage claim area there is no shortage of offers of transportation.
Cost: $10-25 depending on destination.