A good way to start off in Madrid is with churros and chocolate. Chocolateria San Gines has been open since 1894. This place is up a little side street.
On one of the main squares in Gamla Stan is this wonderful cafe. Chokladkoppen is quite easy to spot; just look for the two tall colourful buildings, and the orange one is the one that you want. It is a fantastic cafe to sit and watch the world go by, especially on a hot sunny day.