Madrid is a thrilling city for young people. You can start going to the cinema (oh, don't forget it is in spanish
), going to have some tapas at the city center, then go to pubs and discos till 6-8 in the morning.
So don't let these carcas prevent you from trying everything (save drugs, of course, there is no fun in ending mad or dead). Go to the cinema (you can try original version ones for movies in english), go for tapas, go to have dinner in the vast diversity of restaurants, go to the theater, go to visit monumental and natural spots (during the daylight...), and don't forget visiting pubs and discos, have a couple of drinks and flirt with some nice locals. There is a lot of fun to experience here!
Fernando