The Melbourne Museum, opened in 2000 in a striking modernist building behind the Royal Exhibition Building, is the largest museum in the southern hemisphere. Its permanent collections cover Indigenous Victorian culture (Bunjilaka Aboriginal Cultural Centre), natural history (a full sperm whale skeleton, dinosaurs, an indoor live rainforest), Melbourne's social history including the mounted body of Phar Lap, and a dedicated children's museum.
Allow at least half a day if you want to do justice to more than one wing. Entry is free for Victorian residents and a moderate fee for visitors; the museum is open 364 days a year (closed only on Christmas Day) and combines well with a stop at the Royal Exhibition Building tour next door.