I lived in Löbtau in the 1990s before it was done up, in a coal-heated unrenovated flat not far from the power station, and boy was I pleased to escape from there! I believe it has changed now, though! (Did you know it used to be called "Kuh-Löbtau" as it was outside the city and the locals were considered a bit yokelly?)
I then moved to Pirnaischer Platz in the centre of town (Altstadt) which was lovely and convenient, but you do get Nazi marches going past the door from time to time, complete with armed police ... once looked out the window and they had an elephant walking down the street, so it's not all bad, though! Many of the flats there have been renovated.
Now live in Johannstadt, which is quieter, with more suburban-style roads, wide, relatively flat/cleanish pavements and schools and kindergartens galore, but the buildings are pretty ugly, although the flat does have central heating and everything :-) It's near the Elbe, and it feels safe enough for the kids to go out and play unsupervised.
Here's a random map with the districts:
http://upload.wikimedia.org/wikipedia/comm...emeindungen.pngThe ones which are well connected to the tram network, etc, are those in orange and peach in the centre of the map. There are also good connections up to Klotzsche.
The places to avoid are Gorbitz and Prohlis (rows of huge tower blocks packed with unemployed youths). The Neustadt is popular with young people, but as you walk into Alaunstraße there are usually a bunch of scruffy men dressed in black with studded leather jackets, sitting on the floor with their dogs, so it may not be to everyone's taste. The posh, nice areas are those between the Blue Wonder and Pillnitz, and all up that hill to Weixdorf. Also, Klotsche and the area near the airport. But there are good and bad patches everywhere, it's worth going and having a good look round.