The nearest general aviation airports are San Carlos (KSQL), Palo Alto (KPAO), Oakland (KOAK), and Hayward (KHWD).
Here are some possibilities for rentals:
San Carlos - Rabbit Aviation 650/591-5857
Oakland - Oakland Flyers 510/568-3317, Alameda Aero Club 510/638-6197
Hayward - California Airways 510/887-7686
Palo Alto - Palo Alto flying Club 650/494-7248, Stanford Flying Club 650/858-2200, West Valley Flying Club 650/856-2030
Palo Alto and Hayward are the furthest from San Francisco. I am not positive if all of these businesses provide rentals, but at least they would be able to tell you who on their respective airport could help you.
If you do not have an FAA certificate, you probably will have to go to the local FAA office and get an FAA private license based on your CAA/JAA certificate. I believe the nearest FAA Flight Standards District Office (FSDO) is in Oakland.
Good luck,
Check 6