Come on Board!
If you like history or boats, you should absolutely visit the Australian National Maritime Museum in the Darling Harbour. The admission to permanent exhibits is free, but you should pay the $23 fee (kids pay less) to get access to the vessels docked in the harbour (the replica of Captain Cook's Endeavour and a real submarine).
Endeavour replica is a functioning ship that sailed around Australia in the 1990s. It was amazing to see the size of the boat that had 94 people on board- it was a XVIII Century IKEA when it comes to small space management. Endavour and at the time Lieutenant Cook were sent in 1768 to look for Terra Australis Incognita (unknown southern lands) and observe transit of Venus over Tahiti. Captain Cook sailed up north on the Eastern shore of Australia. In 1770, Endeavour became the first European vessel to make landfall on the east coast of Australia, when Cook landed one of the ship's boats on what is now known as the Botany Bay. On the way back to England, they had a shipwreck at the Grand Barrier Reef, and amazingly were able to fix the ship and continue on. The expedition had a botanist on board and returned to England with numerous species, which all together persuaded the British to start the colonization of Australia.
Next to the Endeavour replica is the HMAS Onslow submarine. Going on a real submarine is very fascinating as it lets you see first hand how small and narrow everything is. Several people had to leave immediately due to claustrophobia- it is awfully tight and small inside, and obviously other than choosing small and tiny crew, they had to select special kind of people who would not go crazy in that claustrophobic environment.
One great permanent exhibit is the Navigators' Exhibit - it talks a lot about how the discovery and settlement of Australia came to be, and about the history of human maritime explorations.