You'll find the primaries in middle-class areas are fine, just as they in the UK. State secondaries are often strictly zoned, the very high-acheiving ones with house prices to match.
Some of these schools, like the private ones, have a winnowing at Year 10 before the VCE starts, and ditch those who will mar the school's ATAR score.
The big difference between state and private is the class size, homogenous intake, fabulous facilities, and their ability to turf out all their problem kids into the state sector. I doubt the teaching is different, as they all train in the same places, and have been told so by a number of ex-private school children.
Catholic schools are not fussy about taking non-Catholics, and the fees are less than non-denominational schools.
Suburbs for living? Central and south east. Northern and western suburbs are, ahem, more challenging.
Do pm me if you want to know more. I live and teach in Melbourne.