Melbourne is perhaps the most European city in Australia. As it is relatively small and manageable, instead of visiting a few monuments here and there, it's better to tour the entire city. We recommend taking the circular tramline. It's free and makes a loop around all of Melbourne (with audio guide). The most essential visits are: crossing Princess Bridge to Queen Victoria Gardens, City Hall, St. Paul's Cathedral, the street of shops called Bourke Street Mall, Parliament and St. Patricks Cathedral.
We recommend stopping midway along the route and enjoying some Australian wine in the Wine Shop, located at 159 Spring Street (in front of the Parliament). Here you can drink a glass of wine for 10 AU$ and snack on something for 7 AU$.