Hola Binky,
I am a flamenco aficionada and was in Madrid last year. I absolutely enjoyed the show at Casa Patas, which is a so called Tablao (bar with dance shows). They are also running a flamenco cultural center and as I understood from their website, they are offering classes, too. Site at
www.casapatas.com (Spanish ans English, although the information about classes is only in Spanish)
The most famous flamenco school in town, if not in Spain, is the Amor de Dios. I have not been there yet (shame!), but they have a good website, too:
www.amordedios.com (Spanish).
If you want to take classes, you might have to enroll for a minimum of one week. (Actually, anything less than that does not really make sense).
With not so much time at hand you should try a Sevillanas class, which is flamenco/andalusian folk dance. It is not hard to learn, even in a short time, and a lot of fun.
The best place to buy flamenco shoes and outfits is Menkes (
www.menkes.es, they have a Madrid branch near Puerta del Sol) or Yebra (right behind the El Corte Ingles department store near Puerta del Sol.
Enjoy your trip! Ole!
Saludos,
la calma