BART station is directly next to the terminal. Exit at Embarcadero station. Most of the BART stations in SF are walking distance of each other. You can actually walk from there to the pier with the Alcatraz ferry if you feel like it. Make sure you get a ticket for Alcatraz ahead of time as it runs out fast. Another option is stop at Montgomery station and taking a cable car to the piers. Also get a city pass you if want to visit multiple things in a day and save some money. http://www.citypass.com/san-francisco
Lyft and Uber tend to have better prices than cabs. I keep both apps around because the difference in price can jump around quite a bit - sometimes up to 2-3x.
However, the ideal transportation if you're visiting is to simply walk around. It's amazing how easy it is to walk everywhere in SF.
For Alcatraz, just grab a lyft/uber to the pier and take the ferry. By the way, if you have time the longer, late evening tours are quite interesting.
If you're going to use a taxi, use the Flywheel app -- it gives you the hail-without-talking-to-someone, see-where-your-car-is-on-a-map, and pay-through-the-app features of Uber/Lyft without surge pricing. Availability is fairly dependent on what part of the city you're in, though.
Might want to be careful with Flywheel. I only used it once, and it defaulted to a 20% gratuity on top of the taxi rate which is usually already higher than Uber/Lyft/Sidecar (unless there's a large surge price). I heard from a friend that the gratuity is adjustable, but the entire experience of Sidecar (the one I almost always use) is far superior in my opinion.
