Melbourne Museum, City Centre, Spencer Street Railway, Flinders Street Station, St. Paul’s Cathedral, Federation Square, National Gallery of Victoria, Town Hall, City Baths, Queen Victoria Market, Royal Exhibition Building, Treasury Building, State Parliament House, Scots and St. Michael's churches, Rialto Towers (with an a 55th floor observation deck), and the Old Melbourne Gaol.
Of the three cities visited on this trip of Australia, Melbourne would be my pick, with its cooler climate than either Sydney and Adelaide and higher deployment of light rail.
Quick Tips:
Resources: MD Travel Health, City of Melbourne, CitySearch Melbourne, Gaytravel, and Google Search=gay melbourne.
Ask your hotel for suggestions, attractions, and any discounts available.
Best Way To Get Around:
Metlink Melbourne provides clean and efficient public transport. Taxis are available, but the meter moves up rather quickly. A car rental is available, but not advised, as walking can be done, depending on how far you’re going.