Madrid is a pretty small city and it's really easy to get around
The Centre is Sol - its really loud and kinda dirty and very touristy. There is shopping there as well as on Gran Via (which is like right next it)
You can find plenty of hostels all over the city (obviously the ones in the centre are more expensive) and the metro is really easy to use and Taxis are cheap. (Bars stay open until 3, Clubs until 5, Metro closes at 1:30 and open again at 6)
You can find out some accommodations on this website:
http://www.madridman.com
Other than hostels there are really cool rooms or apartments available that are a bit pricey but they're really nice looking and if you get your own flat you wont really have to worry about privacy or safety (obviously it's cheaper to get it for a group). I saw some really nice apartments for short term rent in the Plaza Mayor (which is just down the road from Sol and it's also a place every tourist should see)
Popular Clubs to check out:
Pacha -
http://www.pacha-madrid.com/
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Kapital -
http://www.grupo-kapital.com/kapital/
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