Walk down to the Ferry Building for lunch at least one day. There are some pricey places in there but $20 can buy you a very nice meal as long as you avoid Slanted Door and Market Bar. Also, the basement food court down in the Westfield Center (look for Bloomingdales on Mission) has some great under $20 options, including decent Korean and Thai places.

Nightclubs are going to be tough if you are under 21 but I think you have posed a great question that could be very useful for this forum. There's lots of information about what adults and children might like in SF but what should a 20 year old do during the evenings here?