The first thing I do when I move to a new city is open a library account. If you are in Madrid you can head to your closest Biblioteca pública municipal. Last year I went to Iván de Vargas to get my library card. The process is simple; all you need is identification, your local address, zip code and maybe a phone number. Tell them that you want a carné de la biblioteca and they’ll hook you up right there and then.
The library card is valid for all public libraries of Madrid city council, including the Bibliometro ( the reading points in the Madrid metro). It works like any old library card; you can borrow 3 books for 30 days (except reference books), 3 multi-media items (CDs, DVDs, etc.) for 7 days, and 3 magazines for 7 days. Nothing can be renewed.
The card also gives you access to internet/computer stations and Wi-Fi in certain libraries. It is free so take advantage of this excellent resource especially if you live right by one, there are about 17 around the city (find your nearest here). Flyers for all listings can be found at your respective library.