Like mentioned above, stuff to do around the city include Eureka tower, the MCG Tour and National Sports Musem. There's also a decent street-art scene if you're interested in that kinda thing. The trams within the city are now free so that's an easy way to get around (especially the City Circle tram which does loops of the city). The Botanic Gardens are nice and Queen Victoria Market is good for knickknacks. Outside the city, stuff like Phillip Island, Healesville Sanctuary, the Dandenong Ranges and Yarra Valley Wineries are good visits. Again, it comes down to how much time you have.
I grew up in Wellington before moving to Melbourne, so on that front I'd recommend the Te PaPa Museum (always loved that place as a kid) and visiting Mount Victoria. If you get time to go up north, you can visit the Hobbiton set at Matamata.