If you’re staying in an urban area (SF, Berkeley, Oakland) don’t rent a car while you’re here.  If you’re staying in outlying areas, you might use a car to get to and park at a BART station, but parking and traffic in SF is a headache you don’t need.

Instead:

1.  Get a Clipper card, a debit farecard that works on almost all transit systems in the Bay Area.  Ideally, order one online to be mailed to you (takes 5-7 business days), and you can immediately set up Autoload, which reloads your card from a credit card or bank account so you can “set and forget”.  Unused value never expires (though you cannot get it back as cash).  If you don’t have 5-7 days of lead time, you can buy this card at Walgreens drugstores anywhere in the Bay Area, or at the Muni vending machines in the underground stations at Embarcadero, Montgomery, Powell or Civic Center.  Unfortunately you can’t yet buy them at the airports.  Once you have the card, put some money on it at any Add Value machine in the underground stations or at any Walgreens.  Both methods let you use credit cards or cash.  When your trip is over, keep the card for your next visit.  (You’ll surely want to come back.)

Various separate agencies run Bay Area transit, but the Clipper card works on all of them.  BART runs fast trains that connect San Francisco, Oakland, Berkeley, and the outlying suburbs.  Muni runs the buses, trolleys, and “Metro” streetcar/subways in SF.  Other agencies run buses in other counties.

2.  If you have a smart phone (iPhone, Android, …) bookmark NextBus.com, which provides real-time bus arrivals for most Bay Area agencies based on transceivers mounted in the buses.  It uses GPS to detect where you are and give you departures of nearby buses, or you can select a specific route, stop and direction.

3.  Use Google Maps (or the built-in Maps app on the iPhone) to get public transit directions between any two points.  The route and connection info is accurate, but the specific connection times often aren’t because buses may be delayed during peak hours, etc.  The ideal app would combine the directions from Google Maps with bus departure times from NextBus, but as far as I know that app doesn’t exist yet.

4.  Become a member of Zipcar or CityCarShare, both of which have rent-by-the-hour stations all over the Bay Area.  If you’re already a member, everything should just work.

Tips on seeing specific things:

The best way to see the Golden Gate Bridge is to bike across, and take the ferry back from Sausalito or Tiburon.  Bike rentals are available next to the Hyatt Embarcadero Center hotel at the Embarcadero BART station and various locations along the Embarcadero between the Ferry Building and Fishermans Wharf.  The rentals include helmets, locks, and excellent maps.  If you don’t want to bike the bridge, the next best way to see it is to get to the bridge plaza on Golden Gate Transit bus 10 or 70, which is a lot faster than getting there on Muni since the GG Transit buses make very few stops in SF.

The Ferry Building (just outside Embaracadero BART station) was beautifully restored in the early 2000s and now hosts an urban-agriculture farmers market several days/evenings a week.  It is well worth a visit.  Ferries do still sail from here; if the weather is nice, you can joyride the Alameda-Oakland Ferry, which makes a stop at Oakland Jack London Square and another in Alameda before returning to the Ferry Building (whole trip takes about an hour and you get great views of the Bay Bridge and the lay of the land).

Locals tend to steer clear of Fishermans Wharf, but it does have those fascinating sea lions, and other things in its general vicinity are worth visiting, like the Musée Mécanique containing hundreds of really old (some from early 1900s) mechanical amusement devices, the USS Pampanito, Fort Mason Park, the Maritime Museum, the recently-restored Crissy Field, and the Municipal Pier.  Traffic is awful around there, so either bike there from the bike rental at Embarcadero, walk there from the Embarcadero BART (~20 minutes), or take the F-Market aboveground streetcar from in front of the Ferry Building.

The Ferry Building itself, at Embarcadero and Market St. (outside Embarcadero BART and near Transbay bus terminal), is well worth a visit and hosts a local farmers market 4 times a week, at which all products must originate within 150 miles of the building.  Mmmm, fresh.

Outside of SF, Berkeley is well worth a visit, for both the campus and the vaguely funky Telegraph Avenue area just south of the campus’s Sather Gate.  The best way to get there is to take BART to Downtown Berkeley station.

The best way to visit Golden Gate Park or Dolores Park/Mission Dolores is Muni. The Muni “Metro” trams (on rails) serve both—the N-Judah runs the length of the southern edge of Golden Gate Park, the J-Church passes by Dolores Park.  All the tram lines converge on Market St.