Here is a compiled list of places I gathered in my 6 months or so stay in berlin, and also having a berliner as a partner.
hang out places: all with loads of cafes and restaurants
especially near kastanienalle/zionskirche
especially around Oranienstrasse (near SO36)
Galleries and Museums
The East side gallery
for awesome graffiti and all
Museum: I liked the museum of photography (forgot the name but it is in Oranienburger strasse S bahn)
So at the corner where Oranienburger strasse meets Tucholsky Strasse
Also the hambergerbahnhof museum for modern arts is nice
Flea market sundays
Either Mauer park(trendier: nice to just hang without buying stuff) or Boxhagener Platz (cheaper and getting even more trendy)
Pizza place: il casolare: punk pizza place even the italians love
Also for proper german food try Max and Moritz on Oranienstrasse
Sunday brunch in Lychenerstrasse(this is where I lived) or Kruezberg at oranienstrasse/Paul-Linke Ufer oder Maybachufer
Also, there are loads of cheap places near Kreuzberg 36 - near Oranienstrasse which are around 5 euros for a main dish
and Kreuzberg Bergmanstrasse also offers a wide range of cuisines around 10 euros or less.
Don;t forget to check out
Spätzle express : what is spätzle??
**Lazy Caribbean bar
the YAAM - really nice in sunny days
**Drink as much as you want and then pay as much as you want wine bar
Weinerei - all near Zionskirche
even the new york times loved it
Berghain: techono electornic, but the whole idea is to go there, since it's supposed to be this amazing place which opens around 00:00~ 15:00 ... peak is around 4am.
also lots of night life around Warschauerstrasse
but buy a Zitty(events etc guide for Berlin) and see what is happening.
or go to www.zitty.de
but most of all have fun and stay around in the East side...