I take it you mean the Golden Gate Bridge, which is actually "international orange" in color. You can walk or bike across it, or take any number of other tour-related vehicles across it. But the railings on the sides are quite low, so the acrophobic (like me) wouldn't be keen on it even if the Brooklyn Bridge and others don't frighten. The best view of the bridge is from the Marin Headlands on the Pacific side of the Gate, up the little access road.

Don't forget the big part on Saturday night called Magnitude-- if you're into dance parties and whatnot.

Be sure to book your tour tickets for Alcatraz well in advance. They often sell out.

As for Saturday afternoon, just set out walking someplace. There are always gorgeous neighborhoods and beautiful buildings and houses to find. A long-time resident and I constantly walk all over San Francisco, and we always discover new lovely things.

You'll be fine without a car. Plenty of public transport. Consider a MUNI pass valid on the Cable Cars and streetcars running under Market Street. You can take BART (separate system from MUNI) from SFO into the middle of town for about $6 o/w. A cab costs $30-$40.