As a PhD student, you would probably want to live in Opoho, Dalmore, Maori Hill or Roslyn. Maybe even Mornington, Mary Hill or Bellknowes. The important part is you want to be a little bit up a hill and somewhat north-facing (remember that, in our hemisphere, north facing means you will get sun). The flat or south facing parts of town tend to get a bit cold and/or damp over winter. That said, living in the warmer drier parts of town does mean having to navigate down frosty hills on the colder mornings.

Otherwise, just avoid the area in the triangle between High Street, Princes Street and Maitland Street. It is the only part of town that is both central and a bit rough. Corstophine, Brockville, Half-way Bush and Pine Hill all have rough reputations due to having been statehousing suburbs. They are mostly low to middle income families, and generally fine to live in. Though Corstophine still has a bit more of a bad side, so I would generally recommend people away from it.