Vienna or the no-boredom-city
Vienna is indeed world-famous for its architecture, its music and the numerous famous monuments you can visit. So if you decide to spend some time in Vienna, you will have a lot to do. If you don't have time for everything, here are the things I (as a former visitor) can advise you to see.
The one place you absolutely have to visit is the Schönbrunn Palace, the so-called "Little Versailles" (it is a replica of the Versailles castle). It used to be Princess Sissi and her family's home. A complete guided tour exists and is worth the price (10 euros or so for students) if you want to know some interesting facts about the story of this castle. The gardens are also absolutely gorgeous (especcially in summer) and if you are brave enough to climb up the hill in the gardens and reach the top, you will enjoy the view on the whole city...
If you are an art passionate, Vienna is meant for you : music, painting...You will love everything ! Vienna's symbol is obviously Mozart, and the opera house is the place to hear some of his works. Furthermore, the opera house is in itself a beautiful architectural piece of work. About the painting, you can take the best of the numerous museums : the Kunsthistorisches Museum (the museum of art history), for classical paintings, the Bevedere (an anciant palace) where you can see more modern paintings, including the Klimt works and these are only two examples...
But if you are a walker, the Viennese streets will be enough in themselves...As a very dynamic city, the streets are always crowded and make the walk lively! There is always a nice building to look at, classical or modern...
You certainly understand it : a week, even a month won't be enough to visit Vienna...So make the best of your time in Vienna to discover this wonderful city!