Melbourne isn’t Sydney, with its really obvious, beautiful sights. It doesn’t really have any obvious, landmark attractions. It’s more atmosphere and aggregate of experience.
In the CBD: walk around the alleyways for the funky graffiti and nouveau restaurants. Stare up, and admire the Victorian and Art Deco goodness of a confident, rich city.
Walk down Southbank, especially in decent weather (when that happens): it’s a lovely, bustling promenade.
Pop up to Lygon Street, Little Italy, for the gelati and coffee culture; less now for the students from Melbourne Uni, because uni students aren’t as interesting as they used to be.
Go down to the St Kilda pier for a stroll along the beach (such as it is, this is Melbourne after all), and take in the self-conscious bohemia of Fitzroy St and Acland St. You didn’t live here in the 90s, so you won’t feel the stabbing pain in Acland St of what it used to be: a slice of the shtetl turned into deracinated hipsterville. Just enjoy the hipsterville show. If it gets too much, the shtetl is still around the corner in Carlisle St.
Walk through the myriad of public gardens and parks. The Botanical Gardens, the Fitzroy Gardens, Flagstaff.
Go to the ethnic enclaves. Little Greece in Oakleigh; Little Vietnam in Richmond; Little Turkey in Brunswick St, Little Spain in Johnston St. Eat, and eat widely: we have a critical mass of culinary diversity, and culinary innovation.