The Tree streets rule is more applicable to the center of the city. The key areas you want to avoid are the box formed by Chestnut, Cilley, Belmont and Bridge Streets, as well as the Piscataquog neighborhood on the West Side and parts of Notre Dame. There are certainly exceptions to this but you'll have to visit on a block by block basis to get a better read.

Bakersville is a little run down north of Mitchell St and has some public housing. I wouldn't call it dangerous, but the Highlands is a little nicer. I personally don't like the south part of town because South Willow Street easily gets backed up by traffic and lacks character (it's a big retail hub for the region with the mall being there), but there are some good deals on housing over there if you don't mind it. Of course, if you are moving from Boston the traffic and crime here will likely be miniscule in comparison to what you're used to.

Feel free to ask about any specific streets. Manchester is a great little city but the neighborhoods vary widely in appeal.