Claudia, my husband comes from Melbourne and I have quite a lot of friends there with small children. If you tell me more about what you're looking for, I can dig up some more precise recommendations? I am at justsomethingidoatgmaildotcom if that helps. Give me a price range and a distance from the city and I bet I can help.
Melbourne's got quite expensive so it depends what you mean by "fairly central but not hugely expensive". Coburg, for example, will sell you a semi-detached 3BR for $700K, and is about 20 minutes from the city, standard nice respectable suburb, ok schools, parks, nothing special. The leafy suburbs - Hawthorn, Kew, etc (like Unley/Tusmore, etc)) - are really expensive now. North Fitzroy and surrounding suburbs is a mix of trendy little yuppie places and student digs - lots of little trendy cafes and fabulous shopping, but you won't get a place with a decent garden without mortgaging your soul.
Adelaide to Melbourne is quite a leap unless you work in a business where the salary difference will reflect the property price difference. My friends are all thirty--something paraprofessionals, in IT or HR or the Arts, and most of them rent even though they've got little children, because to afford to buy they'd have to move out of the fairly central, interesting, trendy suburbs and into the blander Golden-Grove equivalents, and they're not willing to do so. But it is a nice city. The shopping is definitely better, and it's definitely less provincial!