Hi! I live & study in Vienna, so I can name some places to visit.
Burg - the castle of Vienna. There are some exhibitions, I think the most interesting is the Sissi / Imperial Bedrooms. The "first lady" of the Habsburg Monarchy was an extraordinary women! She was very modern in many ways, her beauty was legendary, and she always felt herself very lonly, as a stranger in the castle.
Schloss Schönbrunn - the summer residence of the Habsburgs, with a wonderful park. Just amazing!
Zoo Schönbrunn - next to the summer residence. If you like animals, dont miss it!
Albertina - the modern arts gallery (I personally didn't liked it very much)
Kunsthistorisches Museum - fine arts. AMASING!
Naturhistorischen Museum - next to Kunsthistorisches M. Life siences, with a lot of interesting temporary exhibitions. Dinos, sceletons, whatever you want. I like it very much.
Technisches Museum - close to Schönbrunn, for technic fans. However, there are some parts, that everybody can like: a CT, music instruments, oldtimers, etc
Michael-places
Imperial - where MJ stayed.
Prater - you can find 2 rides with MJs pictures.
Prater stadium - concerts
If you tell your interests, I can find some more...
Oh, I almost forgot: Kahlenberg: a mountain next to Vienna (the bus drives you to the top), you can see the whole city, a MUST.
I hope this helped you.