Closer in suburbs worth looking at would be Brunswick, Fitzroy/Fitzroy North or Northcote (north), Kew, Richmond or Hawthorn (east), South Yarra, Prahran, St. Kilda or Elwood (south-east), Ascot Vale, Essendon, North Melbourne or Kensington (north-west) and Footscray or Yarraville (west).
South Yarra is probably the most expensive of those (followed by Kew, Prahran or Hawthorn probably), Footscray probably the cheapest.
Some of the closer in suburbs have older public housing towers, which often have a bad reputation, yet they're right near houses selling for a million bucks. A lot of the inner suburbs had a pretty bad reputation a couple of decades ago, but have improved with gentrification.