Well ,well...As i'm from Madrid myself, I have to say I'm impressed with the recommendations people make. One thing I have to insist, even when it has been mentioned before, beware of the pickpockets, they're everywhere (I'm not racist, but lately the country, my lovely country, is full of non-nationals (mind you there is also nationals ) who are making our cities quite un safe, specially morocans, gypsies, etc) (Have nothing against people emigrating for genuine reasons)
If you go shopping, and you want to use your CC/ Laser, make sure you have an photo ID, because you will be asked to show it.
Can't recommend any hotel, because I have free accomodation every time I go there..
As for places to go, I have to recommend one of my favourite "Tapas" place: it's called "Bar Salamanca", it's quite close to the Plaza Mayor, in a street called "Cava Baja" next to a really famous restaurante (Bush's wife was there having lunch with our Queen in their visit to Madrid). Prices there are nothing compare with prices here (went there last december, and spend 18 euros on food and drinks).. Try to avoid turisty places, as you will be ripped off completely..In that same area, there is plenty of small tapas bars that open till about 2.30 am (they are being really strict with closing times in that area lately), but you will be able to find other places opening late than that. One nice night club in the area is called "Palacio de Gaviria", quite close to Sol. Not sure how it's like, haven't been there in a long time. Also, smoking ban was just introduced recently in Spain, but they can chose, and most small places will allow people to smoke, which can be annoying, as spanish people smoke a lot in general (Tobbacco companies are after reducing prices).
If you have time, go to Toledo, it's only an hour from Madrid, and there is trains almost every hour from Atocha (my heart sunk when I passed by there after March 11)
Public transport is quite good and quite cheap. You can get a 10 passes ticket (called MetroBus), that can be used by more than one person, and it's valid for both EMT buses (it's the public bus company) and the Metro for around 6.50 or something like that, otherwise, single tickets are 1.30 I think, but you will see that nobody pays single tickets as most of the people will have the Metrobus or the monthly pass. Bus stops will have information about where the bus is going and where is coming from and some will have info about how long for the bus to arrive!. Metro is quite similar to London, with colour coding, easy to use. Again, be really careful in the Metro, if people push you in the train, and there is not too full, they might be trying to rob your wallet. Also keep your handbag close and where you can see it.
I know the new airport terminal is going to open soon, and the Metro doesn't get there. If you take a taxi, be carefull, they might try to overcharge you. The new terminal is going to be connected by a suttle to the other 3 terminals, and in there there is the Metro.
Sorry if the post is too long, couldn't help. If you need more info, I will be happy to help you. Enjoy!!