Anywhere near a beach will be a long distance from the city. I want to tell you that the beaches aren't real beaches, but coming from the U.K. I don't think you'll mind 
There is williamstown which is 15-20 minutes from the city, though of course it's a bit more expensive. But really nice. Or Port Melbourne and St Kilda, but they are classed as inner city and are expensive. Elwood, Sandringham and Brighton are a few minutes further out, maybe 10-20 minutes to the city, but more expensive again.
Me tone is often mentioned as a really great area, close to the city, close to the bay, and great schools. Might be worth looking at.
Maybe go on realestate.com.au and have a browse through some suburbs, $600-800 per week is about what you would be looking at for a Bay Area. But that might not get you a large house. Do you want a house, with a garden? Or would you be happy with a townhouse type thing with a court yard?
Also definitely consider if you would prefer state or private schools. Private schools can be anywhere from $4000-50,000 per year per child. Depending on if it is a religious/independent or a grammar school. State schools do have some fees but under $500ish per year. Your choice of schooling will really affect your income, and also where you may want to live.
Also the activities you want you family and children will do, you may want to check if the areas you like have the activities that you want. I am sure they would, but nice to check. I think activities may in general cost a little more here too.
Would you have a car? Or be relying on public transport? If relying on public transport then you must live along a train or tram line. You can check where the stations and train lines are on the met link website.