Just a suggestion, social clubs can often be the best way to make friends in a new city. Find a group that shares a common interest & take it from there. This may include language classes, sports, etc.

Easy Expat's guide to Madrid (http://www.easyexpat.com/en/madrid.htm) includes info on Courses for adults:

"The city government, or your local council (the town hall in Madrid) will offer courses in Spanish for immigrants (often for free). They will also give you information on private organizations.

Social Clubs can provide many courses for growth and learning. "Junta" of your area ( xunta de galicia, junta de andalucia...), Universities, employment offices of Inem, provide training and courses for adults. "

 

I should also refer you to a similar thread of another easy expat looking for friends in madrid:
looking-for-friends-madrid-t22634.html

Good luck!