I think your plan to rent for a while before picking a more permanent location is a good plan: there's quite a few different areas might work, and checking them out over the course of a year will give you a chance to get a good feel for them. The different areas have quite different cultures, and it'd be hard to recommend something specific until you've actually been there.
Your budget sounds ok -- as of May 2017, $800 will get you something decent, but by no means excessive.
I am most familiar with the inner west area. Less so with the other side of the city, but there's lots of options there too. No trains on that side, but buses are decent.
Pyrmont might fit? Lots of apartments, a bit of a café scene, and close enough to the city to ride, ferry, light-rail, or walk to work with some great harbourside parks. It can seem a little sterile though. The harbour fringe bits of Glebe might suit too, although there are fewer apartments and more terraces there (and they're getting beyond your budget).
Staying inner west, Surry Hills, Newtown, Chippendale -- further away from the harbour, but good brunch options and well serviced by bus or train.
Another option to think about is that you can catch a ferry to the city quite easily. That gives your a great twice-daily trip on the water, but means you can consider suburbs substantially further from the CBD, including on the northern side of the harbour.
The beaches are also lovely, quite different in general "feel", and the public transport options are more limited (basically buses). Bondi, Bronte ... work your way down the coast.
If you have specific questions, I'm happy to try to answer.