Home to a diverse range of immigrants and plenty of locals, Melbourne is a truly multi-cultural city and the undisputed sporting and arts capital of Australia. Despite its rather unpredictable weather (at least by Australian standards) its population of almost 4 million people love going out and seeing stuff, and if you want to experience Australian football, cricket, ballet, theatre or coffee, you simply have to go to Melbourne.
In 1835 a bunch of Tasmanian property entrepreneurs arrived in Port Phillip Bay and managed to 'buy' 2,000 km² of land from a group of local Aboriginal representatives. The Yarra River provided a good base for settlement, Port Phillip Bay was an excellent harbour, and the area was perfectly situated to become a stop-off point for whaling and commercial ships exploring the east coast of Australia. Farms began to develop around town, impressive buildings sprang up and by 1850 Melbourne was the capital of the new colony of Victoria with a population of 77,000. The future looked bright, but it really started to shine in 1851 when Gold was discovered in hills a couple of days ride away to the northwest.
The population of the city doubled in a year and by 1860 as the gold got deeper and harder to find with a pick and shovel, the prospectors began to move out of the Gold fields and back into the city, pushing the population up to half a million people (making Melbourne considerably larger than the entire population of California, which itself was in the grips of Gold fever). By 1880 Melbourne was the second largest city in the British Empire (behind London) and by the time Australia became a federated nation in 1901, Melbourne was named the (temporary) capital of the country (while they waited for Canberra to be built).
International flights arrive in Melbourne from most countries that fly into Australia. Domestic flights land from all other Australian capital cities, many regional cities and most large regional Victorian towns.
If you're coming by car from Sydney you can take the Hume Highway, which is the most direct route at about nine hours, or the Princes Highway along the south coast, which is longer and still full of pot-holes, but more scenic. The alpine roads via the ACT also offer a scenic diversion through the Victorian mountains. If you're heading in from Adelaide you could take the relatively mundane Western Highway, but if you can spare another half a day the Great Ocean Road is far more rewarding.
Trains run into Melbourne from Sydney, Adelaide and significant sections of rural Victoria and busses cover the same routes with a few variations.
The Spirit of Tasmania Ferry makes daily crossings between Melbourne and Devonport in Tasmania and is the best way to get a car between the two states. If you haven't got a car it's much easier and cheaper to fly across Bass Straight.
Melbourne's public transport system is divided into zones and the one ticket (called a myki) will allow you to travel on virtually any mode of public transport within the zones you pay for.
Trams are the easiest way to get around the CBD and between major suburbs, busses cover most of the gaps and overland trains travel between most of the major suburbs.
Visit Public Transport Victoria for more details.
Watch the ballet, theatre, opera, an outdoor film festival, or anything at all. This is a city of events and socialising.