I'd recommend taking a free walking tour, they go daily from the Brandenburg Gate and visit a lot of the main sites. My guide was really knowledgeable and it's a great way to see the city if you don't have a lot of time. I did another alternative free tour looking at all the street art- there's a lot of really impressive images around the city and it's definitely worth seeing some of these. These tours work on tips so they are really affordable. If you are in Berlin on a Sunday then you should try and go to Mauerpark as there's karaoke and a flea market there then. You don't have to take part, it's just fun to watch (if you do want to go then check online to make sure it is on first)! Also, if you book in advance, you can go into the dome of the Reichstag/Bundestag (Berlin) for free and get some really good views of the city. If you want to go on a night out then Berghain / Panorama Bar is probably the city's most famous club, but it is renowned for being hard to get into. My personal favourite was Weekend Club. It isn't obvious to spot from street level as it's at the top of an office block but it has a roof terrace bar and again you can get good views of the city.
I would recommend Pergamonmuseum on museum island. It is definitely worth seeing if it's still open (due for refurbishment ). It's amazing what they were able to hide during the war. Close to the Zoologischer Garten, there is a pretty big Biergarten that we loved hanging out at. I can't remember the name. Another really cool, really packed Biergarten is across the river from Museum Island. When l say packed, l mean to the hilt, and the stunning view of the island after dark is so cool. Make sure you visit the East Side Gallery, with the remains of the Berlin Wall. We also enjoyed seeing the Hotel Adlon Kempinski Berlin because we loved Liam Neeson saying "is this some kind of joke" in the movie "Unknown". . Here is a post l wrote about Pergamon and pictures to get an idea