The Salamanca district in Madrid is best known for its designer shopping and the futuristic architecture along its Paseo de la Castellana, but the area is also home to some of the city's best small museums. While parts of this district can be quite upscale, there's a lot of diversity and a great place to live on any budget.
To the south you have the Retiro, the city's largest park. Just next to the Retiro you'll find a boulevard, adorned with statues, and some of the most famous fountains in the city, and three of the greatest museums in the world including Museo del Prado.
To get oriented, this page gives a description of the different districts in Madrid:
http://www.timeout.com/madrid/features/172/madrid-area-guide.