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.
The museum is open from 10am till 5pm everyday, apart from Good Friday and Christmas Day. We used public transport to get there, because we know parking can be pretty chaotic in the city.
It was a very good day, and I was pleased that kids aged 3-16 were free. Adults only $10.