Pier 39 is a big tourist destination and you can easily add it to your Alcatraz day. Make sure to book the Alcatraz ticket before you go. Generally they're not available on the day of.

A lot of people enjoy going to Golden Gate Park. The SF Academy of Sciences in Golden Gate Park was recently re done (reopened about two years ago) and is pretty well done with a planetarium, aquarium and an indoor rain forest. The living roof is neat to see as well. While I've never been myself, the Japanese Tea Gardens are close to this museum as well.

This is a free service called 'City Walks' that are given throughout the week. A volunteer takes groups on a walking tour of different parts of the city. I'd recommend looking into that.

The Ferry Building has a great farmers market. I've only been on weekends and don't know if they do them other days of the week.

The Marin Headlands is a great beach just north of the Golden Gate Bridge. It has fantastic views of the city.

Muir Woods is also north of the city and has some wonderful redwoods to explore.

If you have a car (or I'm sure there's tours you can get onto) I highly recommend a trip up to wine country, which is about an hour north of the city. There's no easy way to get to it via public transportation. Most wineries do change a tasting fee that is different at each winery, but is usually around $10. Cline Cellars, in Sonoma, is my favorite, and as a bonus doesn't have a tasting fee. They also have a fairly well down display on the California Missions.

Another nice day trip would be to either Santa Cruz or Monterey. Both nice coastal cities south of San Francisco.

If you have even more time I'd recommend a trip to Yosemite. It's about a four or five hour drive from the city, though, so it would really be an add on rather than something you could do as a day trip.

One thing to note. Mark Twain said 'The coldest winter I ever spent was a summer in San Francisco.' Be prepared. While inland (where I live) we get HOT in the summer, the city itself is kept pretty cool by the ocean. When the fog rolls in it's gets down right chilly! Make sure to pack appropriately and be able to dress in layers. It's a running joke in my family that we all have a ton of 'San Francisco' sweaters because we go into the city from the East Bay under dressed and need to pick up a new sweater to deal with the temperature. :-)