I’ll be in San Francisco over a long weekend. We’ll be staying in the downtown area, and I wish to get some photographs of the Golden Gate Bridge…because no one has ever photographed it 
I assume that the best vantage point is from the Marin Headlands? It seems like public transportation will only get me to the south (San Francisco) side of the bridge. Is there any reasonably priced way (short of walking) to get to the headlands without my own car?
You can find plenty of tour companies that will take you over the bridge, including one where you ride a bike over it, then they load you on a boat for the return trip.
Besides that, there are plenty of good vantage points, like Lincoln Park and Baker Beach on the San Francisco/seaward side, as well as Crissy Field, Marina Green, and Fort Mason, which are closer to city streets.
Marina Green is very close to the Palace of Fine Arts, which is also a photogenic landmark.
As for vantage points, Marin Headlands is where you se the bridge in the foreground and downtown SF in the background. But there are tons of other vantage points available. Here, shamelessly, are my GGB pictures, all taken from San Francisco (save a handful from Treasure Island,Angel Island and the East Bay). If you’re interested in any of these angles, let me know.