Hola! I just got back from Madrid and here are my opinions on where to live, how much to expect to pay.
A studio will run you anywhere from 550 to 800 euros depending on size and location. A one bedroom can go for more, probably starting at around 700 euros. A deposit is always required and expenses for utilities are not very bad. Cost of living in Madrid is, in my experience, low compared to the U.S. at least. In fact, I was in Mexico City over Christmas right after coming home from Madrid and I thought groceries were exhorbitant after being in Spain.
Some of the nicer neighborhoods are Salamanca, anywhere near Serrano and to a lesser extent, Chamberi. Malasa~na is good for young people, as is Arguelles (lots of students). I lived in Tetuan and I loved it. I wouldn't live anywhere else since it is right on the number 1 line and has lots of ethnic restaurants and grocery stores. I like diversity. Not everyone does.
Good luck!