[fabadam]

I've sent you a fairly extensive PM about Amsterdam itself.
If you want to look further away, there's Amstelveen to the south, which has very good public transport to Amsterdam. Amstelveen is quite expensive mostly however.

There's a lot of places in varying quality if you're willing to look further away. You would probably need a car for most of those places if you want to go anywhere.
I have a personal bias for Haarlem, because I live there, but it's a very nice small city.

If you just want to live in Amsterdam itself, you can do without a car.

Seeing what you're asking in this post, I think your chances are best in the areas just outside the center: Oud-Zuid (de Pijp especially), Oost/Watergraafsmeer (near Oosterpark), and Oud-West, Baarsjes and Bos en Lommer. The latter is a bit rundown in some places, but hardly unsafe.
(There's always some safety risk in a city like Amsterdam, so you need to take care to lock your doors well and stuff, and if you have a bike, you're need a honking big chain for it.)