Since I don't know Bilbao, I'll offer you nothing more than the official website
http://www.bilbaoturismo.net/BilbaoTurismo/en/tourists
and the (always wonderful) Wikitravel site:
http://wikitravel.org/en/Bilbao
The 'hostal' I stayed in while in Madrid was lovely. It was up on the 4th floor and consisted of only about half a dozen rooms, with a really friendly & helpful proprietor. I had a shared bathroom but I never saw anyone else using the shared facilities anyway, and they were always immaculately clean. The furniture in my room was old but perfectly functional, with an exceptionally comfortable bed. The air-conditioning (operated by a remote control unit) was excellent. There was free wi-fi and a telly too. The location was on a side street just off the Gran Via, with a Metro station very close by. (i.e. about as central and well-located as it's possible to get). The price for all of that (for a single room) was just €22 per night. HIGHLY recommended!
http://hostalmachin.com/ING/index.htm
Madrid is great for art-lovers. The Prado is unmissable:
https://www.museodelprado.es/en/
The modern art museum is well worth a look:
http://www.museoreinasofia.es/en
My favourite art museum, however, was that dedicated to the work of Joaquín Sorolla:
http://museosorolla.mcu.es/
The Railway Museum is interesting too:
http://www.museodelferrocarril.org/en/principal.asp
as is the Royal Botanic Garden:
http://www.rjb.csic.es/jardinbotanico/jardin/
For far more attractions, see Madrid's official website
http://turismomadrid.es/en/
and Wikitravel
http://wikitravel.org/en/Madrid
Madrid has an excellent, modern, air-conditioned Metro system; it's easy and cheap to get around:
https://www.metromadrid.es/en/viaja_en_metro/tarifas/abonos/contenido02.html
There are loads of cheap places to get meals, including this small chain (with 5 centrally-located restaurants): Amazing value!
http://www.allucaneat.es/en/index
(If you fancy something classier you won't have far to look either. Madrid is packed full of restaurants and bars!).