Prague has become one of Europe’s favorite cities, but I suggest you also try to visit more of the fascinating Czech Republic. Karlovy Vary is a veritable outdoor museum of art and architecture with buildings from the 18th century with Baroque churches and palaces, to the newly restored late 19th-century Russian Orthodox church which would look more at home in Moscow. Most of the buildings date from the late 19th and early 20th century, many in Art Nouveau, especially the surrounding hillside villas. Marianske Lazne has the longest and most beautiful spa-iron arcade with frescoed ceilings and even an enclosed bandstand, but it is unconnected to Karlovy Vary by public transportation.
There are many other day trips across Europe that allow you to visit multiple great sites that you can’t with public transportation. Here’s a short list:
From Paris: Vaux le Vicomte, and Fontainebleau, a palace older and more completely furnished than Versailles, are only a few miles apart, but there is no public transportation between the two.
From Madrid: The stupendous city of Segovia with its iconic majestic castle (of Ferdinand and Isabelle fame) is not connected by public transportation to the perfectly-preserved Roman aqueduct.
From London: There are three world-class sites almost impossible to do together by public transportation in a single day. One of Britain’s greatest medieval cathedrals, Salisbury, the mysterious pre-historic Stonehenge, and the city of Bath are not connected.