There's many ways to get around san francisco by public transportation, but if you're going to travel out of the city, you might want to rent a car. If you want to go to the amusement park (paramount's Great America), it's a little drive, and you should check to make sure it's open (it's not open all year round, i think opens in march/april- not sure though)...
As for the Palace of Fine Arts, I used to work there- the Palace of Fine Arts is just the name of the area, there is no actual art museum there- the Exploratorium is there (a hands-on science museum thing)... there are a few art museums, but they are more downtown...
I would totally go across the golden gate bridge... just hope for no fog! lol
I would check out the Academy of Sciences (has an aquarium, planetarium, etc...) I would of course check out Pier 39/ Fisherman's Warf Area- lotsa cool shops and good restaurants...
If you have any Q's, just let me know and I'll try to help