Prague Castle - you will need ~2 hours; in general, walking around the Old Town is great to soak up the atmosphere and grab a beer at U Černého vola (don't expect good service - expect good beer!) Petrin Tower - great view of the city and nice walk! Strahov Monastery - worth a visit, especially the library, place where most of my family got married/baptized Malá Strana - Old town quarter, great to walk around and soak up the atmosphere - then you can easily cross Charles Bridge Prague Astronomical Clock - also good to get on top of the clock tower! National Theatre - if there's a play on, the interior is really nice Ss. Cyril and Methodius Cathedral - where the paratroopers that assassinated Heidrich hid away and died, really nicely done now Vyšehrad - beautiful cathedral, lovely park, most famous cemetery (with the likes of Dvorak, Smetana buried there) National Memorial on Vítkov Hill - off the beaten path but a great museum of Communism and view of Prague from another angle
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Liberec - a small town with a nice city centre - not worth more than an afternoon Český ráj - a beautiful national park definitely worth an afternoon hike Krkonoše National Park - another beautiful park, where you might want to spend a night or two
Tabor - nice town, worth a stop if along the way, but not a must see; one of the main towns in Hussite movement in 15th century Kutna Hora - medieval town with beautiful cathedral; historically significant because of it's silver mines - worth a stop, not to spend much time in Český Krumlov - one of the most picturesque towns in the country, worth spending the night in Šumava National Park - beautiful natural reserve with a lot of hills, forests, rivers and swamps - great for 1-2 nights In Moravia: the biggest wine and Slivovice region in the country: Brno - second biggest city in the country, worth a stop along the way but not to spend much time in Pálava Protected Landscape Area - where a lot of wine is from, many towns and villages around there are very nicely maintained and the region is worth 1-2 nights of stay for sure. Some highlights would be Mikulov, Valtice and Lednice. From South Moravia (towns mentioned above), Vienna is only 1 hour away