Munimadrid is a public program to help Madrileños learn and use new technology. The city has set-up 21 learning centers where residents can connect to the internet and ask knowledgable instructors about computers. The Munimadrid website contains lots of very basic information on everything from buying a computer to making a webpage or playing computer games. The information is accompanies by great anthropomorphic computer cartoons like the ones in this post (here's a Flickr photostream so you can see them all). On the Munimadrid site, you can also download Sun's StarOffice 7 for free...after filing a petition. This is still Spain after all.
El País contains a lengthy tech news supplement every Thursday. In addition to many informative interviews and articles, the Ciberpais supplement contains practical links and tutorials. Most of the supplement can be found on the El País website. EP3 is a Friday supplement containing tons of edgy articles on music, art, sex, literature and culture. Even if you don't speak a word of Spanish, you can enjoy the well-designed EP3 webpage.
If you're looking for free wireless in Madrid, check out this list. For a reliable connection and people who won't bug you, I highly recommend Giangrossi (#5 on list). Also, our neighborhood hangout, El Naranja, now has free Wi-Fi.