While Melbourne’s public transport system is often maligned by its residents (especially the aging train infrastructure), the Metlink system that connects the city and suburbs with trains, trams and buses is relatively well-planned and a reasonable value for the money. However, well-planned and well-executed are two very different things. Despite the infrastructure being in place, public transport is not always reliable. There are regular cancellations, delays or hold ups on many train lines, and overcrowded trams are known to occasionally cause chaos at peak travel times.
If you plan on living in the Melbourne CBD or inner suburbs, a car is not essential. Many people choose to live without one, particularly because of the difficulty finding parking in inner city areas if your accommodation doesn’t include a parking space. However, if you have any intentions to travel beyond the built up inner city suburbs (and you should, Australia’s countryside is beautiful), then you may find a car useful, as public transport can be frustrating when trying to complete a big shop or transport large objects. If you plan to only use a car infrequently, there is an abundance of car-hire / rental agencies that can provide reasonable daily or weekly rentals.
Air travel within Australia is extremely well organised, convenient, and competitive. In recent years several budget carriers have appeared on the scene to provide commuter trips between Melbourne and other Australian cities, and most of them have multiple daily cheap flights that reliably leave on time, and can be booked with little notice.