San Francisco Airport is about 25 minutes from downtown by taxi, which will cost you about $45. The quickest option is to catch an Uber or Lyft, but make sure to walk to the Departures terminal; these services are not allowed to pick up passengers from the Arrivals level. The major public transportation system, Bay Area Rapid Transit (BART), goes directly from the airport to downtown and across the bridge to the East Bay.
BART trains, Muni trains, buses, cable cars, and trams traverse the city. Most rides cost about $2, and you can buy tickets and passes in any of the stations. If you're getting on a bus, Muni train, or cable car at a street stop, you can pay in cash as long as you provide exact change. Taxis are readily available, as are rideshare services like Uber, Lyft, and Flywheel, all of which you can use through apps you can download on your smartphone. It's also fairly easy to rent bicycles from the Bay Area Bike Share program, which has terminals throughout the downtown area.