The city of Berlin is a Bundesland (a German state) in itself, much like Hamburg and Bremen. However, it sits in the middle of the Brandenburg Bundesland, therefore the area is generally referred to as Berlin-Brandenburg. The capital of Brandenburg is Potsdam, easily accessible via S-Bahn (make sure you have a travel card to zone C). The city is a UNESCO World Heritage Site. On a sunny day, the visit to Potsdam is very nice. Best if you rent a bike at the station because all the castles and palaces are far from each other. Don’t go if the weather is not nice or if you are tired.
To get there, simply take the S-Bahn. Just make sure your ticket is valid to zone C. The journey from Zoo Bahnhof takes around 30 minutes and trains depart regularly.