There are so many different factors at play here, the major one being price. Also how close to the Melbourne CBD you need to be. I would recommend Canterbury, Camberwell or Kew area if you can afford it because of how close to great schools they are. But again, it's such an open ended question that every suburb in Melbourne would probably suit you to one degree or another.
It depends what kind of lifestyle you are looking for, in the suburbs area...usually near the south east is ideal for a good social and quieter environment. I am in Narre Warren South and work in Berwick, there are plenty of schools, shops and public transport here with a low crime rate!
Generally the eastern side of Melbourne is good for a family with kids - Malvern, Prahran, City, Hawthorn, Camberwell etc are all good places to live. However so are other places - Burwood, Burwood East, Heidelberg, etc are also good even when not so close to city. Places like City Wharf are great also.
Moonee Ponds / Essendon. It's great from a public transport aspect, shopping and cafe culture are A +and not as judgemental as Lygon or Chapel streets and close to the city and north / western suburbs. Lots of parks for kids and families to enjoy. Can be a tad expensive though with renovated cottages and houses.