- the beautiful Retiro park, great squares like Plaza Mayor and Plaza de Cibeles, big avenues like Gran Via
- the Royal Palace, the Royal theatre, the national library, the temple of Debod
- great museums like Prado, Museo Nacional centro de art Reina Sofia, Thyssen-Bornemisza museum, the archaeological museum etc
- watch a match of Real Madrid at Santiago Bernabeu stadium
- do a day trip to
Toledo or
Segovia. Both of them are so picturesque! Check also El Escorial.
I liked to walk the downtown of the city where many of the basic attractions are. The center is always busy and crowded but there’s always a metro/subway station to use if you go to go further or return back to your hotel. It runs 6:00-2:00am something very useful after some drinks :) Just buy a metrocard(6.70) for 10 metro rides.
What I like most in Madrid is going to the Tapa bars and have beer and jamon! There is at least one in every block so they ideal for resting, socializing and meeting with locals. Late at night, there is always a good concert, many blues and jazz clubs but also rock pubs, theatres and cinema.