Hire a bike and get out to Potsdam so pretty!!! Also, the Berlin Philharmoniker do free lunchtime concerts on Tuesdays which are superb!
Kreuzberg is ace- das gift is there for some Mogwai malt. East side gallery is free and well worth a look. TV tower gives you a birds eye view.
Go and see the house in no man’s land!
Food: There is a real down-homey brewery in the now-very-fancy Rosa Luxumburg Platz called Marcus Brau. It’s like eating in someone’s German grandparents’ living room and they brew their own beer. So worth it. There’s TONS of good Vietnamese restaurants, especially in KBerg; Kebabs are everywhere, cheap and magical and you should eat them for the other two meals of the day. KBerg is good for food but if you really want choice go to Friedrichshain where every second doorway is a restaurant.
The wall memorial (near Brunnenstraße) is awesome, markets on the weekend are great (Mauer Park and Boxhagenerplatz are the reliable ones), if you want to stay away from tourists don’t even bother with the centre of the city. The Stasi museum in FShain is awesome/terrifying and in the original Stasi offices. If you want a terrifying tour go on the Berliner Unterwelt tour of the nuclear bunker built into a UBahn station; probably the most amazing thing I did in all the years we lived there.