Today we took our kids to the Melbourne Museum. They found it really fun. The museum is in Carlton, close to Melbourne's CBD.
I would recommend taking children aged 5 and up to the museum as it's educational and more about learning things, so younger kids may get bored. They have lots of cool exhibits, including cars and horses (they also had a replica of Phar Lap)
There is lots of hands on things to do; my kids enjoyed the mind and vision exhibit because it was interactive and there was lots of stuff to check out on how the mind works. My boys favourite part was the indoor rain forest. It was cool because it felt like we were outdoors, but we weren't.