what kind of things do you like to do?
art...architecture...sports...beer 😉...photography...?
personally, i just like watching people. if it were me (well, this is one thing i normally do when we are in spain) is just find a nice little tavern/cafe, grab a table and watch. you'll probably find many people doing the same and gossiping in madrid...
you might be able to catch a bullfight if you are brave enough to watch - i'm not myself...
v