The first thing that comes to mind is Luna Park in St. Kilda, about 4 miles (2.5 km) south of downtown Melbourne. It’s a nice little amusement park that the kids might enjoy. You didn’t say how old they are, but if they like science, the Scienceworks Museum in nearby Spotswood would be fun for them. They might also enjoy the Melbourne Museum just northeast of downtown.
For you, there are numerous botanical gardens and you will probably enjoy shopping at the Queen Victoria Market and the two historic shopping arcades, the Block and the Royal. There are also two beautiful cathedrals; St. Paul’s and St. Patrick’s.
I hope that’s helpful. There are many, many other cultural opportunities there. Melbourne is a beautiful, friendly city with superb public transportation. I had a great time there and I’m sure you will too.