The first day I was there my friend and I spent the whole day just walking around the main city centre, exploring, and eating cake (Sachertorte to be more specific - an Austrian kind of cake) in a typical Vienna 'cafe house'. Where the waiters are rude, you wait for ages to be served, the tea comes without milk, and when you ask for some you will get it by the time you've already finished your tea. But that's what it has to be like in Vienna, I've been told. ;)
Definitely was a nice day, and a good way to get a first impression of the city, though.