Metros of Madrid

After using the one of the euro rail passes to get here, I went exploring Madrid - I never realised it would be so huge!  Every place I need to get a metro which I don't like to much, I prefer cities where I can see it by foot and get a taste for the atmosphere.  However, being so big, it does have advantages over the smaller cities...

So much to do, I orginally planned to stay a week but think will be staying a little longer to get a taste for life here.  potentially i could get an internship here (i can see it entertaining me for 10 months easily) but I need to settle a bit here I think...get in a routine, to have a proper idea of it.

Any recommendations of places to see would be great ...more in the way of local bars and cinemas.  Like I say in my description, I'm on an economic budget and so tourist attractions are minimum.  Also, I want a resident's point of view if the city, not a tourist's - i would after all be living here....