It's really a toss up in selecting Melbourne or Sydney, they both being quite reasonable cities, both having some great residential areas, much average and some areas not so good.
Melbourne is somewhat regarded as more the sporting capital though it's more from having things like the Melbourne Cup, Tennis Open, the MCG, and the Gran Prix or they did have it whereas Sydney has just as high participation and possibly more by way of triathlons, the Harbour and Coastal surf beaches against Melbournes Bay sid beaches and Sydney has slightly warmer winters and more humid summers.
As for suburbs, Sydney rentals are probably a bit more expensive and it can be more a case of get what you can and it'll depend on what you're prepared to pay and how fat to commute etc.
Better suburb areas in Sydney would be Northern Beaches and NW, Easturn Suburbs very expensive and less so in South, and west or southwest, you are heading for the wild west.
Melbourne is a bit the same except there are no northern beaches but replace that with southern suburbs around the bay and forget the North to NW around to the west, again wild west territories, NE around through East to South East being reasonable.
Melbourne if anything will have a greater selection of inner suburban areas that are not so bad.
But using that general guide, check out a site like
www.domain.com.au and use maps as a resource, google earth also being good to get a good idea of what it's like on the ground, Sydney traffic a bit more chaotic.
For schooling, I think you'll find all schools are on an Australian schedule though you could do a search for if there are any international schools on northern hemisphere schedules.