Madrid is awesome and there are so many things you can do in 2 days. Visit 'La Latina' and try some tapas hopping plus a 'caña' (beer) and stroll among locals. Or just sit and relax at 'Plaza Mayor' with a 'sangria jar'. Another square you can visit: Santa Ana where there are some bars, a theater and restaurants.
Take a walk by 'La Gran Via', 'Callao' and 'Sol' Lots of stores, bars and restaurants.
Another day, you can take it for museums/art. From Atocha station you can walk to Reina Sofia museum (which it's my favorite) and then walk by Paseo del Prado and you'll pass by the Botanical Garden, El Prado museum (it's free on Sunday and I think after 6pm any other day), and then you'll see Neptune fountain in front of the Ritz. If keep walking by Paseo del Prado you'll get to 'Cibeles'. From there, you can visit El Retiro park passing by 'Puerta de Alcala' first.
If have time and like football/soccer, don't forget to visit Bernabeu Stadium. Another option, visit Plaza de Oriente, the square by the Royal Palace.
Have fun and enjoy Madrid! about 10 years ago