Money saving suggestions
goldstar.com has discounted tickets for many events in San Francisco, including the walking architectural tour of SF, the ghost hunting tour, the Beach Blanket Babylon show (a city institution since the 1970s), the Wicked musical, etc.
Public transport: There's BART (a regional train), MUNI (city-wide bus, light rail/metro, electric bus, and the cable cars), and Caltrain (connects to suburbs). MUNI may be the most convenient option for you. You can get a map of the city with all the MUNI routes shown, and buy a 3 or a 10 day pass that will include the cable cars, which is a pretty good deal. The cable cars are very fun on the steep hills - don't miss them! If you ride cable cars boarding lines are shorter away from the terminal stops, and ride on the back platform next to the break person. They are always eager to share fun stories.
The best way to see Alcatraz is to take the night tour. You should book it well in advance, as the availability is very limited, and the prices are much higher if you walk-up to the pier:
http://www.alcatrazcruises.com/website/buy-tickets.aspx
The California Academy of Sciences museum in the Golden Gate park can be expensive, but on Thursday nights between 7 and 10 p.m. the tickets are discounted to $12 and they have music and drinks too. Get the tickets online to save time. As you get in first thing you should do is to get a token for the planetarium. It's the same show that's running right now in New York Natural history museum though, so if you saw that you can skip the planetarium.
DeYoung museum across the plaza from the Academy of Sciences has one of the best collections of the Oceania artefacts in the world, and it's not something many museum display anyway, so it's worth seeing. Their temporary exhibits can also be interesting.
Japanese tea garden next door to DeYoung is very nice. It's $5 entry, but the entrance is free on Mon, Wed, Fri between 9 and 10 a.m. It closes at 4:45p.m. in winter (till the end of Feb,), and at 6:00 p.m. starting in March.
Asian Art museum is medium-sized, reasonably priced, not very crowded, and easily acessible. Highly recommend.
Exploratorium is a science museum aimed at children first, but I still loved it in my 30ies.
In terms of entertainment, Beach Blanket Babylon is pretty famous. It's a musical review where Snow White travels the world in search of a prince to marry, and encounters different characters. The specific characters are updated constantly to parody the latest political and cultural news. A San Francisco original!
Aunt Charlie's Lounge has a drag show on Wed, Fri and Sat at 10 p.m. that has a very unique atmosphere: blue collar public, performers from young to very old (the lady with a liquid spine must be in her 70s), very friendly and relaxed if you are a straight person in a gay place. It's a small neighborhood bar.
North Beach and Polk Gulch neighborhoods offer a lot of more mainstream nightspots.
You can walk over the Golden Gate bridge and back, but get a pair of earplugs and be prepared to breathe stinky car exhausts.
The Castro theater is a restored movie palace with an interesting movie program (can be found online). Definitely worth visiting if you have time. Before the show there's a guy who plays a Wurlitzer organ (but not before the second part of a double-header). The last song he plays is always "San Francisco, open your golden gate" from the movie about the earthquake, that's how you know the movie is about to start!
The food is very diverse and can be had both very expensively and very cheap. Consult yelp.com for eateries near where you will be staying.
Other than that just walk around the town. Financial center, Chinatown, North Beach and Telegraph hill, Haight-Ashbury, Twin Peaks, Dolores Park and Mission Dolores, Golden Gate park, Lands End, Ocean Beach and Cliff House/Sutro baths ruins are just some of very walkable neighborhoods, each with a different feel and architecture. They all can be accessed by MUNI.