As far as cultural areas go, you've got your hip, water-side precincts like the Docklands (downtown) and St. Kilda, replete with esplanades, theme parks, gardens, and historic eating and nightlife spots. Enjoy these spots at night when the lights shine out over the water. Of course, there's also the Central Business District, where the iconic Flinders Street Railway Station and the unsightly modern hub of Federation Square rub shoulders with public markets, plenty of museums and famous public buildings. These include the Queen Victoria Market, the Police Museum, and the highest residential building in the southern hemisphere, the Eureka Tower. Take some time to waltz around here before trekking out of the city on your next great adventure.
Then again, there are other minor landmarks in other parts of the city; the massive Melbourne Museum and the Old Gaol (a jail where the night-time tours lend quite a creepy element to the proceedings) in Carlton, the Australian-rules stadiums of Etihad in the Docklands and the Melbourne Cricket Grounds just east of city center are just a few.
Although not bursting with the obvious, bonk-you-over-the-head history of your typical European destination, Melbourne provides a high-end reckoning of modern culture and should do for anyone looking for a few important cultural relics and plenty of popular, bustling avenues.