While Leipzig is worthy of a short break itself, a little over an hour by train, it is also a managable day trip from Berlin. Plenty to do day and night, it's a city full of history, proud of it's leading role in the 1989 revolution. There's an interesting array of museums, including the former Stasi headquarters, now documenting the history of the Stasi. It also has some great nightlife with fab cafes and bars to kick back with a beer or cocktail.
Potsdam is a gem of a city, right on Berlin's doorstep. Just 30 minutues away by train and you are whisked away to world heritage sites and plenty more, from the gorgeous Schloss Sanssouci and parkland, the mini 'Brandenburg gate' which opens into the delightful shopping street Brandenburger Strasse, the Dutch Quarter or Park Babelberg. Potsdam is a not to be missed day trip for anyone visiting Berlin.
To the south west of Berlin are the Wannsee lakes and the great Strandbad Wannsee, the largest inland lido in Europe. This is one of my favourite places to visit when in Berlin. If the weather's good what could be better than a day at the 'beach', having a swim and watching the boats sail by? Who would believe this is just a tube journey away from central Berlin. A short walk through the Grundewald from Nicholassee S-bahn and the lake opens up before your eyes!
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