For anything not covered, I highly recommend http://www.stilinberlin.de/ for pretty much all of their recs.
Michelberger Hotel - Berlin version of the Ace Hotel. Killer location close to the Spree river and East Side Gallery with all the Friedrichshain clubs walking distance to the north and chill bars, cafes, restaurants in Kreuzberg across the bridge to the south.
Airbnb - look for flats near the U8, which people say is the L train equivalent.
Kreuzberg- This is kind of the Williamsburg of Berlin - becoming gentrified but still pretty cool. The graffiti around here is awesome as well. Spend some time here. I stayed in this area the first time I visited and loved it. It's a bit further out from central Berlin, but close to the big clubs.
HardWax - Dope record store (Paul-Lincke-Ufer 44a 3rd floor, 2nd courtyard, door A.)You have to go up some stairs in what appears to be abandoned building. If you think you're in the wrong place.. it's probably right :)
Blackjump Mural / Floating Giants - again the graffitti here is some of the best in the world. You'll find Blackjump Mural at Falckensteinstrasse 48. There are some Banksy works around here as well.
Friedrichshain - This area is near Kreuzberg, you should do both in the same day. Just walk across the bridge after you see all the big murals. Really the most important thing here is the East Side Gallery. It's touristy but definitely worth seeing, the art is beautiful.
Mitte - The soho/west village area. Great restaurants and shops. Amazing area to stay in as well. Go to Cocolo Ramen, it's fantastic (Gipsstraße 3, 10119 Berlin, Germany).
Museum Island - Berlin has a lot of history. Terrible history mostly, but history nonetheless. You could spend years in their museums. I believe you can get a student pass that will get you into all of the museums for a flat fee.
ViktoriaPark - probably better in the summer time to hang out in, theres a waterfall at the top that runs down towards street level, just young crowd, pretty landscape and fun.
- Klunkerkranich - rooftop bar on top of a mall in Neukolln - you have to take the elevator up and walk through the parking lot area to access it
- Natanja & Heinrich - best cocktail bar with Parisian vibes also in Neukolln
- Melody Nelson - also a sweet cocktail bar but in Mitte
- Vin Aqua Vin - good wine bar
- Hopfenreich - best beer bar
- Geist im Glas - good local bar
- Five Elephant - nice cafe
- Lokal - seasonal high German cuisine - not brats and potatoes
- Azzam - Turkish, best hummus and stuff
- Markthalle Neun - indoor street food market on Thursday evenings
- Mustafa's - most famous kebab shop but always a huge line
- Burgermeister - arguably Berlin's best burger
Other things worth mentioning that you are bound to find anyways..
- The Jewish Museum (closer to PotsdamerPlatz) and the Pergamonmuseum are both very good. Jewish Museum is really really amazing, extremely sad given the history.
- Brandenburg Gate - big gate, cool story.
- The Reichstag (German parliament) - You have to make an appointment to climb to the top, It's free to do and offers a nice view of the city. Good way to get your bearings and the audio history guide on the way up is pretty cool. My suggestion is to book it early in the morning when it opens and you can make a reservation in the restaurant on top to have breakfast..
- Holocaust memorial - this monument can be interpreted in a lot of different ways. It's sad, but really well done.
- Humboldt University - maybe better than Oxford.. just maybe.
- Berliner Dom - it's a dome, duh
- Flohmarkt im Mauerpark - flea market on Sunday afternoon situated at the old east/west wall border.
- Volkspark Humboldthain - uphill walk for sweet views and trees
- Sammlung Boros - a converted air raid shelter that houses an amazing private collection of contemporary installation art. You have to book a spot on the guided tour in advance.
Nightlife / Clubs:
Berlin revolves around electronic music. It's actually less about the venue and more about the DJs. I suggest using Resident Advisor for events in Berlin for the days you're there.
Their guide to Berlin nightlife is great as well. Trust all their recommendations. Watergate, Sysiphos, Arena are all good.
And if you really want to go to Berghain.. wear black (think rick owens style), no visible logos, don't speak english in line, keep your phone hidden, and be prepared to take everything off when you're inside ;)