If you’re learning Spanish in Madrid, then it’s quite likely that you will probably need to buy some books. You may also be interested in participating in a language exchange, where you can speak Spanish with a native speaker and vicce versa.

The Casa del Libro offers a great selection of Spanish Language Study Books and is probably where most people purchase their Spanish books from. If you want English Language books you can also find a small section here. Here are the two main locations (the 2nd is my preferred one):
Locations: Gran Vía, 29 | Metro: Gran Vía; Maestro Victoria, 3 | Metro: Sol or Callao

FNAC is one huge book/electronic/music department store and also has a good selection of Spanish language books.
Location: Preciados, 28 | Metro: Callao

Crisol is another place where you can find Spanish Language Books. They have a number of shops across the city.
Locations: Marqués de Casa Riera, 2 | Metro: Banco de España; Serrano, 24 | Metro: Serrano

Petra’s International Bookshop is also a good place to pick up second-hand books. It is also a great place to meet up with people for conversation exchanges. They are open from 11am to 9pm.
Location: C/ Campomanes, 13. Metro Opera – Santo Domingo

O´Neills´s Pub is also a good place to go for language exchanges on Tuesday evenings. They start at 10pm and you should ask for David.
Location: C/ Príncipe, 12 . Metro Sevilla/Sol.

Cafe Madrid is the place to go for language exchanges on Wednesday evenings (starting at 9pm) – you should ask for Fran.
Location: C/ Escalinata | Metro Opera

Wings & Beer have been meeting every Tuesday for around 2 years to practice languages, meet people, organize parties, trips, etc. Location: C/ Martín de los Heros, 4 | Metro Banco de España

Lola at Estudio Sampere has also told me that the school organises exchange parties once a month and they have a list of people who are interested in language exchanges.