South-eastern suburbs are way to go. South Yarra, Toorak, Prahran. St.Kilda is in the same area, it is right next to the sea. Chapel street and Toorak road cross this area, lots of city life, night life, shopping, eating out, pretty much the "melrose place" of the city. But it's still a great residential area. Downtown is close, several met lines go through this area so you dont have to wait on your train, you get on any east bound, they all have a station in South Yarra.
If you are on a budget, then keep going south-east, Ripponlea, Armadale and so on, it gets more quiet and "residential". Or Brighton Beach, if money is not a concern, is one of my favorites
Or Williamstown Beach on the other side of the bay is a great place to live.
North-western suburbs are more blue-collarish, if you will (Essendon area). I'd say eastern suburbs Hawthorn, Camberwell,... are better choice (never lived there tought).