You should visit the Museumsufer. There are more than ten Museums and Art Galleries on both sides of the Main river. See also http://de.wikipedia.org/wiki/Museumsufer (German only)
If you like electronic music, visit the Robert Johnson Club in Offenbach. It's one of the best electronic clubs in Germany.
In the summer, go to Goetheturm and enjoy the view over the whole city.
Or go to Lohrberg for a panoramic view from a different angle.
Visit the public platform of the Main Tower (central Frankfurt), check out one of the many locations around Zeil, visit the great Opera (Willy-Brand-Platz)