There are TONS of possibilities. Some that I've enjoyed for a stay of several days:
Berlin
Istanbul
Lyon
Madrid
Barcelona
Lisbon
Prague
Budapest
Palermo
Naples
Taormina
Frankfurt
Stockholm
Copenhagen
Bergen
Edinburgh
Note that these are all quite different, not only in themselves, but in the kind of experience. For instance, Berlin and Istanbul are very big cities, where you would be seeing lots of museums and attractions during your three days (on the move a lot), while Taormina is more of a Bruges-type place, where there are few "attractions" but there's a great atmosphere and ambiance, and the visit is about relaxing and soaking in the vibe (and beauty). That doesn't mean there aren't ways to relax in Berlin or Istanbul, but you may feel more "pressure" to keep going to see everything if you're only there a short time.
Also, for some of these you would feel you had "done" them in 3-4 days, while for others you would feel you had just scratched the surface. Of course, which ones are which depend on you, and how you react to each place, as much as they do on the more objective factors like size and number of attractions. For instance, Berlin has many days of major sights, but if you don't like it, that doesn't matter - you'll be ready to leave in a few days. Whereas you can "see" Taormina in a relatively short time, but experiencing it takes longer (I was there 3 days and didn't want to leave, just because it was so nice, not because I still had a list of attractions to see).
And, of course, some of these are better at certain times of year, so when you're traveling will influence which ones you will want to go to.
Some other places that I didn't enjoy quite as much as others do, but that certainly have enough going on to keep you busy for a few days, are Vienna, Munich, and Krakow. Then there's Oslo, St. Petersburg, Moscow - the list really is endless.
How to start? Read about destinations, ask your friends what they've enjoyed (and not enjoyed), and watch movies and videos (Rick's are available on this site, here: http://www.ricksteves.com/watch-read-listen/video/tv-show
Have fun choosing!