A good place to look for an apartment is Segundamano, which is essentially a cross between classified ads and Craigslist. There are a number of hostels all over Madrid - here's a good source for info. For the more adventurous, there's also Couch Surfing.
This great post at Lonely Planet contains a wealth of advice on travelling around Western Europe. To find cheap flights leaving from Madrid-Barajas, the best search engines are Trabber, eDreams and Viajar. There are more, but I’ve had the most luck with these three. You might also what to see the specials for each major airline: Iberia, Spanair, and Air Europa. AirMadrid and Vueling are two low cost airlines. Look in the free papers handed out by the Metro- there are often super cheap specials. If you are looking to travel by train, Seat 61 is a great guide.
Phone & Internet
You will you probably have the smoothest experience with Telefonica, even though it costs a little bit more. Because Telefonica maintains most of the phone lines, it's easier to get problems fixed. (You will pay Telefonia around 17 euros per month for line maintenance no matter who you choose). Like I mentioned the other day, there are several places where you can get free wireless: see this list and El Naranja. With cellphones there are more options and lots of special deals.
This is the best article I’ve seen on finding a job in Spain. Unless you’re from the EU, it can be hard to get working papers. Your best bet is to get a job through an employer in your country with an office here to arrange a transfer. If you take classes at a university and have a student visa, you can work a certain number of hours. Many foreigners here teach English - both legally and under the table. If this is what you want to do, ask around the English-speaking circles (J&J Books is a good place to start). Also, Escape Artist is a good resource for finding jobs across the world.
See this previous post for some good Spanish sites. The BBC page is awesome. Ultralingua is a great resource for translation and conjugation. Alpha Dictionary has a lot of specialized dictionaries (i.e. Spanish economics words).
Getting Around & Doing Stuff
Munimadrid has info on everything. Their tourism guide is quite nice. Here is an interactive map of the Metro . If you are looking to get around, find a street, or see movie listing, try La Netro (movies are usually dubbed - look for V.O., i.e. original language).
Good resources to find out what's going on are Le Cool (Barcelona and Madrid), Top Madrid, Looc, and Pacha's magazine. Also, Pixel y Dixel and Elastico and two blogs that often discuss interesting places and events. The best sites in English are Mad About Madrid and In Madrid.