Madrid has two globally prominent art museums in the centre of the city. El Museo del Prado (The Prado) is truly spectacular, with arguably the best collection of Spanish art in the world. The art includes a wide range of artists from the 12th - 19th century. This is definitely worth a lengthy visit.
Reina Sofia, named after Queen Sofia of Spain, is a great testimony to modern art in Spain. It was founded in the name of the reigning Queen in 1992. It holds a wonderful collection of art by Salvador Dali and Pablo Picasso and so much more. The closest staion is Atocha and it is definietly worth another lengthy visit.
The City is certainly not all about museums, for those who love gardens and outdoor life, there are parks. The most famous and prominent is the central park of Parque Retiro. Here ther are tree-lined paths, cafes, bars, a large lakes, ornaments and nice gardens. Given Madrid attractive dry climate, this can be a great place to relax.
There are many great areas to go out in Madrid. In the area aorund Banco De Espana, there are some upmarket bars. One of these bars is Ramses bar, they have great interior design and are quite an experience. If bars in the day is your interest, why not finish a lovely walk in park with a quick champagne in Ritz Garden Bar. For shopping, Gran Via, Sol and in particular the El Corte Ingles (Department Store Spanish Chain) may be of great interest.
Above: Plaza Mayor with thanks to Wikipedia from author Sebastian Dubiel
For Plaza and sitting out in large square, Plaza Mayor and Plaza Santa Ana are very popular.
All the same food is extremely important in Spain and a night of tapas is most common. La Latina, Chueca, Plaza Santa Ana all possess speciality tapas bars. Be sure to try Bacalao (Cod), Gambas Ajillo (Garlic Prawns), Jamon Iberico (Ham of the highest quality), Lomo Iberico (Pork Tenderloin in a type of Ham/Chrorizo format) and much more. Madrid is a city where wondering around can lead to you stumbling on a great find.