No-one has mentioned nice places to live such as around the New Town, Broughton St, Rodney St, Forth St, off Picardy Place, Barony St, E Claremont St, E London St, Broughton Rd, Eyre Place, Gayfield, top of Leith Walk, start of London Rd, Regent Tce, Leopold Place, Stockbridge, Inverleith, Canonmills, all north of Princes St!! Probably a bit expensive but then again renting in Edinburgh is very expensive, and has always been the case.
Areas to be avoided perhaps would be The Fort (Leith), Craigmillar/Niddrie (unless you were born there although a friend did have a house on Hay Drive which was very nice) Bingham, Magdalene. Parts of the Inch are fine.
My advice would be to avoid old mining villages or towns as well, it is rather hard to get accepted as an outsider, such as Tranent, Prestonpans or Newcraighall.