OMG! I think I have a new favorite European city. Vienna, you impressed me. Cool friendly people, fantastic architecture, nice shopping, great galleries. I will definitely come back soon.
A couple of things that should be on your to-do-list (in addition to the compulsory visit to the Gustav Klimt museum and enjoying an opera or two):
Go to Cafe Europa for Sunday brunch. Friendly staff and good food.
Café Sperl: Drink a wiener melange and eat the traditional cake, Sachertorte, at Hitler’s favourite café. OK, so they don’t advertise this, but I have this from a very trustworthy source. An old, majestic place- with free Wifi!