Palacio Gaviria (club) - Calle Arenal 9, very close to Puerta del Sol. It used to be a palace that was transformed to a club. Thursdays and Fridays are great. On Thursdays they have student parties which are fun. Don't come too late since there are lines across the block (around 11pm I think should be ok).
Sesamo (Bar)- Calle Principe 7, there is a red sign on top and you go down the stairs to a basement with a piano player and quotes of famous peoplein the walls. This is a MUST.
Except for that just go from one bar to the other, people outside will invite you for a chupito which is a chaser with your drink. Good Times!
Another place is the Hawaiian bar - on Plaza Santa Anna.
Bars - Near Puerta del Sol, there are 3 streets that form a triangle. When the bear in Puerta del Sol is at your back, you turn left at Calle Carera de San Jeronimo. It is paralel to Calle Alcala and go up 7 mins until you hit Plaza Canalejas. 2 streets on your right: Principe and Calle de la Cruz. Take Principe until you hit Plaza Santa Anna, and just circle it and go down and the other side to form that triangle. You'll see.