Madrid’s public transportation system is absolutely fabulous. I highly recommend choosing the area you live in based on the metro stops surrounding it, and the difference it makes if you have a long commute (as people in the Auxiliares program usually do). I chose to live in Malasaña and although this wasn’t the ideal location for the shortest commute (I could have lived closer to Moncloa, the bus station that connected me to my bus to my pueblo) living in Malasaña did set me up in the center of town, right next to Tribunal (line 1) and close enough to line 3 that although it took me 20 minutes to reach Moncloa, it also only took me 20 minutes to get south to La Latina or Lavapiés.
If you will be taking the metro a lot, it is recommended that you purchase an abono transporte, which is essentially a metro pass that you put money on every month, the amount of money depending on how far out of the city you have to commute. The center is zone A1, then there is zone A2, A3, B1, B2, and so on. I ended up having to go out to zone C1 everyday, making my monthly payment a heart wrenching 86€! If you are under the tender age of 23, you get a youth discount, but alas, I was not so fortunate.