Also check out the barrio (means neighborhood) Chueca - close to Plaza Cibeles - it is the gay neighborhood and has good restaurants, trendy shops, bakeries, art galleries, etc... the "in" neighborhood in Madrid...
Also barrio La Latina - good food at night, interesting area, nice architecture...
And WATCH YOUR WALLET - lots of pickpockets in Madrid - many are groups of Romanian girls who try to distract you while others pick your pocket, reach in purses, etc... Be suspicious of anyone who just comes up to you in the street and starts talking to you.