Vienna

Vienna is just one of the great cities that you can easily reach from Prague.  Probably the best way to get there is to take the train, but there are also buses, and you can also hire a car and driver service to take you there.  The train from Prague takes just under 5 hours, and it is also direct and makes very few stops.

Vienna, once the capital of the Austro-Hungarian Empire (and therefor the "capital" of the Czech lands for more than 4 centuries), is famous for many things, from music to art to pastries and cafes.  ,  
Prague-Main-Train-Station
Prague's Main Train Station (Hlavni Nadrazi) Interior, where most of your trips to nearby destinations will start.

Dresden

Dresden is another international destination that you can get to easily from Prague.  Just over two hours by train, you can easily make it a day trip.  What's more, the train ride itself is one of the most beautiful in Europe, tracing the sandstone cliffs of "Czech-Swizterland" and "Saxony-Switzerland."

Having been badly damaged in World War II, Dresden has been almost fully rebuilt, and culturally it is emerging from the darkness that was behind the Iron Curtain.  Here you can visit the fantastic Old Masters art collection in the Zwinger, take in an opera, and enjoy great grilled wurst from street stands. 
Prague-Vienna
Viennese Baroque Statues

Berlin

Prague-Berlin
The Brandenburg Gate in Berlin
The same train that takes you to Dresden from Prague continues on to the sprawling modern metropolis of Berlin.  This vibrant city is old and new, and spans the former east and the west.  This mix makes for a fascinating experience of a re-emerging city.  From classical and high art to Trabant rides and secret police tours, Berlin has something for everyone.