I'm going to assume (similar to Rob) that you're initially looking for suburbs which are well connected by public transport.
Australian cities are layed out in a similar manner to the US, with a central business core or downtown area (which we call the CBD or Central Business District), and suburbs fanning out from there.
Public transport is generally radial (designed to get you in and out from the suburbs), with little facility for go across suburbs, and is of a reasonable standard close to the CBD getting progressively worse the further you go out.
Melbourne's CBD is situated at the northern tip of Port Phillip Bay, a large natural well-sheltered harbour. Watersports are popular in the bay during the summer.
Suburbs to consider:
- MELBOURNE - the city centre itself. As a starting point, this may not be a bad option. Rents aren't as high as you might think, because not that many people want to live right in the city. Lots of student accommodation and small apartments. Because of Melbourne radial public transport system, being in the city places you at the heart of al transport links so you can easily explore other suburbs from there. You also have lots of eating and entertaining options. Downside, during the week its full of workers.
*. INNER SOUTH EAST
These suburbs are generally up-market and close to the water. Popular with urban professionals and those who like to be near the water and can afford to live further in.
ST KILDA - Right in the heart of Melbourne's busiest entertainment district, great beaches and parks, lots of eateries and drinkeries. Downside, can be a bit noisy. Rents are pricey and places are often cramped.
ELWOOD - Next suburb along from St Kilda. Still pricey, but much quieter, green and leafy, very pleasant indeed and still close to the action.
SOUTH YARRA - Closer to the city than St Kilda, an upscale neighbourhood close to the Botanical Gardens and other parks. Very close to the city centre and arts precinct.
PORT MELBOURNE - Used to be a dilapitated port district, now renovated and brand spanking new. Right on the beach and well services by cafes. Close to ALBERT PARK and other nice suburbs.
ALBERT PARK - Named after the eponymous park. It's where the Grand Prix is held each year. Lots of entertainment. Also close to other nearby suburbs like St Kilda, Port Melbourne, etc.
DOCKLANDS - Right next to the city. Looks like it landed yesterday from outer space. totally out of character with the rest of Melbourne, but suits some, especially if you like "modern living". Expensive rents, and you still need to get into the city to go anywhere else.
*. INNER NORTH
The northern suburbs are Melbourne's more bohemian hangout and attract a young, arty and diverse crowd. Tattoed hipsters, multi-ethnic communities, trendy arthouse cafes and veggie gardens. Uber-cosmopolitan.
FITZROY - this suburb is so hipster, you could put a roof on it and call it a theme park. Locals refers to themselves (only somewhat tongue-in-cheek) as "Fitzroyalty". Lots of cafes, restaurants, clubs, funky bookstores, converted warehouses and expensive apartments. Here trendy hipsters rub shoulders with recent immigrants and refugees.
BRUNSWICK and NORTHCOTE - Fitzroy's more affordable but still quite trendy cousins.
*. THE EAST
Green, quiet, upper-crust, leafy suburbs. Nice house and parks. Big box stores and sprawling shopping centres (malls).
*. THE WEST
Traditionally blue collar and industrial. Unless you're considering WILLIAMSTOWN or YARRAVILLE (right next to the city) or perhaps FOOTSCRAY, stay away.
*. THE OUTER SOUTH EAST
A long line of suburbs hugging the bay. Popular with anyone who basically wants to be by the water. If you like boats and beaches and don't want to pay the exorbitant prices closer in, this is for you. Much nicer than the OUTER SOUTH WEST.
*. OUTER SOUTH WEST
If you can't afford the Outer South East, this may be the go for you. Lots of new estates littered with McMansions.
*. OUTER NORTH
Melbourne major growth corridor. Cheap real estate and brand spanking new subdivisions. Pizza and gangster mansions, but perfectly fine if you just want to live there and within reach of the Inner North. Mostly interesting of you want to live in a nice, new place with cheap rent.
*. THE NORTH EAST AND OUTER EAST
The hills. Literally. Melbourne is flat as a board, until you hit the hills. Then you're in the hills. Enjoy your hills. Generally a bit up-market, and you start seeing nicer houses again, but certainly not cosmopolitan. Great if you plan to own a motorbike.