Free: The National Gallery of Victoria, the State Library, St Patrick's and St Paul's Cathedrals, and Victoria Market. The Melbourne Town Hall has free tours. Victoria Police Museum admission is a gold coin donation. ACMI has free and ticketed exhibitions. Plus there's nearly always some form of free entertainment on in Federation Square including buskers, music, screenings, markets and cultural events.
My favourite places are the beautiful gardens around Melbourne, such as the National Rhododendron Gardens, The William Ricketts Sanctuary, the Royal Botanic Gardens in Cranbroune and so many more in the Dandenong Ranges.
The numerous gardens, ACMI, monthly free boat tours (courtesy of Port Melbourne council), Southgate free cinema, State library of Victoria free film screenings, free tours of State Library of Voctira, Parliament House, Town Hall and so on.