Posted: Fri Apr 11, 2008 5:15 am-
I haven't lived here for very long either, but currently I have a place in Marchmont (adjacent to Bruntsfield), which is home to a mix of students and families with young children. It's quite pleasant, isn't too loud most of the time, features a Neighbourhood Watch program (or so the signs say, anyway), has several elementary-schools, and is within roughly a half-hour's walking-distance of pretty much everywhere in the city-centre. The University isn't far away, but it's separated from Marchmont by the Meadows, a large park (albeit one that's rumoured to be a bit unsafe after dark).
I can't be of much help when it comes to the suburbs, though they might be worth a good look as well.
Definitely avoid any place near the Cowgate, which is dark, sketchy, and infested with nightclubs (and, accordingly, drunk students). There are also students all over Nicolson Street/Clerk Street.
Let me know if you have any more questions!