Just a point of clarification re Southbank, Southland etc.

Southgate is a shopping and restaurant complex on the South Bank of the Yarra, adjacent to the Casino.

Southland is a shopping centre some 20km south of Melbourne City.

You don't need or want to go to Southland.

I would agree with the sights most of the other posters have suggested.
The city is famous for its laneways with shops cafes and bars, walking around will allow you to get the feel of these, and the general architecture.
The National Gallery (Art Gallery) has two branches. In Flinders Street is the Australian collection, and in St Kilda Road the International collection. Either are worth visiting.