I am a Mexican Interaction Designer, I rent my own flat in Madrid when away for holidays I've lived in London for a while studying, my boyfriend is a British English teacher who just recently moved here. We have a small chihuahua called nana, she is very calm and tidy.
My flat is small but comfy and handy with everything necessary within 5 min walk distance, the Prado, Thyssen and Reina Sofia Museums are all within 2 blocks around my flat, very good Market Anton Martín, the Art Decó Cinema Doré is a must, with foreign films, if you want some nature there is an amazing amazing wall garden in this new cultural space called the Caixa Forum that also has great exhibitions and is free all the time also just right in front of the Botanical Garden and Prado. If you are not that much in to art, there is also Media-lab Prado, which is an open lab for people who wants to develop technology related projects, everything is for free and open there, really great people.
Cafés and bars, there are all kind of places in my area, from jazz to flamenco, amazing galician restaurants with loads of seafood, vermut from the tab is very common, I can help you out with all necessary info, map, free hours for the museums, cheap breakfast places, list of favorite local, hope you have a nice time in this city.
Oh! I almost forgot. Atocha train station is just right 3 blocks away, it has a train service to the airport that is 2.50 and a direct bus service as well that is 5€, I don't know about the first one as it's brand new but the bus takes between 30 and 45 min to get there.