Firstly, the metro system is excellent. Pretty much every part of the city is covered, certainly every attraction seems to be located within close proximity to a station.
Eat at both Lateral (map) on Calle Fuencarral and Taberna del Chato (map), they are delicious.
The rooftop terrace of Círculo de Bellas Artes (map) has spectacular views on Madrid. Admission is roughly €3.
Chocolateria San Gines (map) has been serving chocolate con churros since 1894, and they’re phenomenal.
If thinking about going on a tour of any of the football stadiums I’d recommend Estadio Santiago Bernabéu (map) - home of Real Madrid). The stadium is well worth visiting, the self-guided tour should be around €20. Nonetheless Estadio Vicente Calderón (map) is a great stadium even if it is showing its age. It reminds me of the majority of stadiums in Italy.