OK, so you want to get to and from downtown Palo Alto by public transit, on a Friday evening, Friday night, Saturday all day. The location is near the Palo Alto CalTrain station, which is also a major bus hub.

The primary popular mode of transit to the area is CalTrain, with fixed-rail service to and from San Jose and San Francisco and points inbetween. At Milpitas you can connect with BART to get to most of southern SF and the East Bay, with bus, rail and ferry connections from there to Vallejo, Sacramento, and beyond. CalTrain has frequent weekday commute service, but is much less frequent on Saturday, and doesn't run past midnight. BART is 20-minutes-frequent, but takes a while to reach downtown SF, and also doesn't run past midnight.

On weekdays the Dumbarton Express ("DB") bus runs from Union City BART, but there is no service on weekends.

SamTrans (the Peninsula bus service, reaching as far as SF, with Golden Gate Transit connections to the North Bay) and (Silicon) Valley Transit Agency buses from the San Jose area, down to Santa Cruz, intersect at the Palo Alto station.

There is an all-night bus service connecting Palo Alto with San Francisco, connecting to the East Bay and beyond.

The station also features a BikeStation, with secure bike parking and possibly bicycle rentals.

For transit planning assistance, dial 511 from any phone or use the online planner at:
http://transit.511.org/
More specifically, the trip planner can be found here: http://transit.511.org/tripplanner/index.asp

Note that for CalTrain you must buy your ticket before boarding, so allow extra time; the penalties for being onboard without a ticket are now quite stiff.