As the previous poster said, there is not really an area to avoid in Dresden. I think a tourist probably does not want to stay in one of the districts that's full of apartment blocks from the communist era (like Gorbitz or Prohlis), but even these are not 'to avoid'.
The old town (Altstadt) would be most central for most tourist highlights. The New Town (Neustadt) is split in two very different parts: Inner Neustadt (around Königstraße and Hauptstraße), which is relatively quiet, a more upscale neighbourhood with more expensive shops, Baroque townhouses, nice pedestrian zone; and the Outer Neustadt (north of Albertplatz), which is the vibrant nightlife district, mostly younger (many students) people, lots of restaurants (internation/ethnic, too), bars, small theatres, clubs etc. And then there are the beautiful 19th century residential areas like Blasewitz, Striesen, Strehlen, Loschwitz, Weißer Hirsch ... which are also excellent choices if you want it a bit quieter, away from the tourists, also with nice little shops and restaurants, and only a short tram ride from the old town.