Fitzroy is on the north side. Brunswick Street, Fitzroy is a nice shopping strip. Lots of good cafes, and interesting one-off, i.e. not chain store, shops. We lived in North Carlton - good quality housing, good transport links and 5 minutes drive from Fitzroy. It has a very good secondary school (Princes Hill). Not too into Collingwood as a place to stay - too many council flats and can be a bit dangerous at night. It has an interesting shopping strip though (Smith Street, Collingwood). Northcote, just to the north east of Fitzroy, is a good place to live. I had a friend who lived on Northcote Hill. She had brilliant views over towards the city. Going a bit further north, I'm not into Thornbury, but Ivanhoe is really nice. Other than Ivanhoe, I wouldn't recommend living north of Bell Street (9 km north of the city, running east/west.
The Eastern suburbs are all pretty expensive. Balwyn has transport links that go through to Fitzroy, and has the best state school in Melbourne, but you wouldn't get much change out of $500K there.
For child care, we used one in Collingwood, which I think from memory was council run and opposite a small park with a few swings and a slide. The name of it escapes me, but it was very good. For sure though, your hospital colleagues will be well versed on good ones and ones to avoid.
For info on melbourne suburbs, try this site:
www.cityhobo.com, select suburbs and then Melbourne. Then just grab a map (so you know where they are :-) and check out each suburb listed. They are not all listed, though a lot of the inner city ones are.
If you fancy living at the beach and still being inner city, I recommend St Kilda.