Go on an All Day Tour with Brewer's Berlin
Whether you are spending a day in Berlin or several years, this is an absolute must. It's the best way to get to know the city, its history, and a lot more! The guides not only tell you about Berlin and it's very turbulent past, they also have many sometimes amusing, sometimes poignant anecdotes to share. During the tour they point out sites that are worth visiting, as well as restaurants, live-music bars and much more and are full of pratical information. As far as I know all their guides are brilliant, and Terry and Preston are the best Berlin has to offer.

Rent a bike and spend the day cycling around Potsdam
City Rad is located just outside the Potsdam S-bahn station. For about 12 euros you can rent a bike for the whole day (until 8pm). They also provide a not-perfectly-accurate-but-good-enough map of the area with an advised circuit of about 15 kms (the area is flat, so this is far less daunting than it may sound). This route takes you through the castle gardens and past many places of interest (including a beer garden for a well deserved break). Bring a guide book for information about the town and its sites. If you're interested in seeing the inside of the different castles, get a 12 euro (9 euro for students) two-day pass, which gives you acces to all the castles except the central part of Sans-Soucis as well as Charlottenburg and other sites.

Visit Sachsenhausen Concentration Camp
This is obviously not an amusing outing, but is definitly worth seeing. Catch the S1 to Oranienburg, then bus number 804 to the camp. Entry is free, and they provide an excellent english audio guide (I cannot remember if there is a small fee for the audio guide, but I think not). The tour can supposedly be done in about 2 hours, but I don't see how this is possible. I would recommend taking a whole day to see this as there is also a museum on the site. Possibly bring a picnic as there is little in the way of refreshments in the immediate vicinity of the camp.

Take a walk along Landwehrkanal
The Landwehrkanal is about 10 kms long and runs east-west from Schlesisches Tor to Charlottenburg, through the southern part of the city. Though it is possible to walk or cycle its whole length, the most pleasant area for a stroll is from Hallesches Tor to Kottbusser Tor. There are many cafes and restaurants along its banks, especially east of Prinzenstrasse, including some on very picturesque boats (though these are not the cheapest). This is a lovely place relax and maybe read a book in the grass on a nice day.

Listen to jazz at Zosch's on a Wednesday (or Thursday) night
Zosch's is situated in Tucholskystrasse, off Oranienburgerstrasse, just next to the S1/S2 Oranienburgerstrasse stop. Every Wednesday night, around 10:30 pm, a band plays some jazz in the basement of this as yet unrenovated building in former East Berlin. The basement, like many of its kind, was used as an air-raid shelter during the war, as attests the door at the back of the stage. There is no entrance fee, drinks are cheap, and the band usually plays until about 1 am. Just remember that the S+U-bahns stop running at about midnight, so find out which night-bus to catch if you're planning on staying until the end. On Wednesdays it's "LaFootCreole" with a very classic (80-100 year old) repertoire. Thursdays the JazzFormation "JustFriends" plays a bit more more modern Jazz (from 40 to 80 years ago).

Spend a day at the Zoological Gardens
If you're at all interested in animals, this is definitely a whole day outing as the zoo is huge and there is a lot to see. There are many births here in spring as the zoo has a successful breeding program, which says a lot about the well-being of the animals. The most well-known attractions include the pandas, the gorillas and the lions, but what I enjoyed most was the nocturnal house, the feeding of the seals, the penguins and the hippos (which can be seen underwater through glass cages). It is possible to get either a simple ticket or one combined with an entry to the aquarium.

Have supper on board a cruise boat on the Spree
I did not have the opportunity to try this out, so I do not know if it is as good as it sounds, but it definitely sounds great! On Fridays and Saturdays, between May and October, they offer a 2 and a half hour sightseeing cruise along the Spree (with description brochures in english) and an evening meal for less than 20 euros. The tour is also possible without the meal, and several other tours are available.

Go for a swim in the Badeschiff
This is a swimming pool built out of a converted barge which floats on (or in) the River Spree, between the Oberbaum and the Elsen Bridge. There is a beach and pier area for sunbathing. In the evenings the Badeschiff turns into a bar and outdoors club. Swimming is permitted until about midnight. There is a 3 euro entry fee (more on special occasions).

Stroll though the flea markets on a Sunday
Berlin is full of bustling flea markets, where you can find just about anything. If you've just moved into an empty appartment, this is a good place to look for some really cheap deals. There is a market every Sunday on Boxhagener Platz, one at Ostbahnhoff Station and another near Schoeneberg, to name just a few. Bargaining is more than acceptable, it is the expected thing to do!

Watch a movie at the Sony Centre in Potsdamer Platz
A choice of recent movies with original sound tracks. Ticket prices are reduced on Tuesdays, though it is then often necessary to buy the tickets early. This does not apply to student card holders, as the reductions are equivalent.

Make the most of Berlin's many Museums and Memorials
Museum Island is a World Cultural Heritage Site, and is as beautiful as it is fascinating. Five of Berlins most famous museums are found here. Some are still undergoing restoration, but the others are well worth visiting. My personnal favorite is the Pergamon, which houses the Collection of Classical Antiquities and much more. If you only see one museum while in Berlin, this should be the one! Admission is free from 6 to 10 pm on Thursdays, though there is a small fee for the audio guide, which is indispensible.
But that's not all! The Jewish Museum covers 2000 years of German-Jewish life, though the accent is of course on the Holocaust. The building itself is a fascinating modern structure, full of artistic expression and symbolism.
The Museum at Checkpoint Charlie is dedicated to the story of the wall and the courage and ingeniosity of those who crossed it and their helpers. On display are cars, suitcases, hot-air balloons and other devices used in daring escapes. This museum is extremely interesting, though it is worth noting that it was opened well before the wall came down and tells its story from a western point of view.
A block west of Checkpoint Charlie, down Zimmerstrasse, you will find the Topography of Terrors where the Gestapo headquarters used to stand. The building was destroyed, but the cellars have been uncovered and now house an exhibition, which is mainly in german, but a free english audio-guide is available in exchange for an ID card or equivalent.
The Memorial to the Murdered Jews of Europe lies just south of the Brandenburg Gate and consists of a rather controversial Field of Stelae, which is permanently accessible, and an extremely moving Information Center, open until 8 pm. Admission to both is free.
There are many more interesting museums and memorials in Berlin. Unfortunately not all of them have english audio-guides, and even fewer are translated.
Student cards are generally accepted everywhere, even if they are not ISIC.

Try some of Berlin's cafes, restaurants, beer gardens and cocktail bars
The areas around Oranienburgerstrasse in Mitte, Simon-Dachstrasse in Friedrichshain and Moeckernstrasse in Kreuzberg are very lively, especially in the evenings and on weekends. They offer plenty of places to go for a drink or a meal, but here are a few of my favorites:
The Tacheles was once a luxury department store but was badly damaged during the war. On a nice day you can get excellent mojitos from the outside bar and in the evenings the cafe Zapata often hosts live music and dancing.
Across the road from the Tacheles, at the corner of Auguststrasse, there is a van that sells chips (ie: french fries) and saussages. The van doesn't look like much, but their chips, served with the sauce of your choice, are devine! (I would definitely advise staying well clear of the saussages though!)
If you're looking for a more substantial meal, a few minutes away on Linienstrasse, where this street meets Oranienburgerstrasse, you will find Gambrinus, an excelent restaurant that serves typically german cuisine.
The DK, in Simon-Dachstrasse, is a great place to go for a cocktail. Their drinks are excellent, they have a large selection and the prices are reasonable.
The Balzac coffee, Starbuck's rival, makes one of the best hot chocolates I have ever tasted! I highly recommend their dark chocolate with cream... don't forget the cream, never mind about those few extra calories! They also have delicious cakes.
Many cafes serve brunch on Sundays, which starts around 10 am and can last until 4! Usually, for a fixed price of between 5 and 8 euros, you can stay as long as you like and help yourself to whatever is on the buffet. This may include cold meats, eggs, bacon, saussages, cheeses, spreads, salades, fruit, cakes, mousses and more... Drinks are not included. My favorite place is the FARGO cafe on Boxhagener platz (77 Grunburgerstrasse), where you can get a delicious brunch for 6,20 euros, with a view on the rather lively flea market.

Temporary accomodation in Berlin
If you are looking for accomodation for visiting friends or family, have a look at what Berlin Flats has to offer. The appartements are nicely situated, in perfect condition and the agency is pleasant to deal with.