if you have a health care care or pension card you can get into the museum for free- otherwise its a small charge
but be warned- pomeii is on at the moment and the lines are long, but if you dont have to pay, you can get in quick!

The fire station museum is also a good one and they do free displays one day a week. Kids get to see the fire hoses etc in use and they can climb all over the fire trucks afterwards and try on the fireys jackets etc. Its located just off victoria pde!

Fitzroy gardens and the fairy garden, botanical gardens are suppose to be ace as well

there are markets at southbank on a sunday and they are good as well

go have a look around at docklands- sometimes free things there as well
there is a website that has a list of whats on in the city- its called that's melbourne