Berlin

Potsdam
is the capital of the surrounding federal state of Brandenburg, not far southwest of Berlin, and makes a perfect day trip. Especially the park of Sanssouci, a world heritage site with its great famous palaces, is worth a visit. You can get there with the S-Bahn S7 or Regional-Bahn RE1 to the station Potsdam Hauptbahnhof or Park Sanssouci fare zone C. It takes about half an hour from Berlin Hauptbahnhof or Friedrichstraße.
Sachsenhausen
is in outer Oranienburg, a quiet suburb housing the remains of one of the Nazi concentration camps on German soil. There's also a small palace in the center of Oranienburg.

The Müritz lake region to the north is a national park with a few hundred lakes.

To the south, Dresden is 2.5 hrs & Leipzig is about one hour by train.

The beautiful Baltic seashore e.g. Usedom is near enough for a day trip by train.

The Spreewald is a protected UNESCO biosphere reserve. It includes low-lying areas in which the river Spree meanders in thousands of small waterways through meadows and forests. It is a beautiful, unique landscape about one hour south of Berlin and well worth a day trip or a weekend trip to relax from the buzzing city life.

Frankfurt an der Oder
on the Polish border is within easy reach.

Lutherstadt Wittenberg is about 1.5 hours south of Berlin. Schlosskirche was the church where Martin Luther hung his Theses. Across the street from there is a visitor's center with great information. Great city to tour and one can easily explore on foot.

The Raststaette Grunewald at the S-Bahn station Nikolassee is a good spot for hitching if you're heading south or west.

The Polish border is just some 90km to the east of Berlin, therefore it might be interesting to do a trip to:

Szczecin
Stettin in Poland is about two and a half hours by train.
Poznań
Posen in Poland is three hours by train.
Warsaw
Warschau in Poland is five and a half hours by train and tickets are available in SparDay/SparNight tariff for €29 or €39. Check also Polski Bus bus link. (http://www.polskibus.com/en)