Not far from the city centre, you can find Luna Park, which is a small theme park with rides that are fun for all the family. The park is open throughout the year, and very popular on the weekends.
If you have rented a car in Australia, then you can drive the Great Ocean Road, which runs along the top of cliffs, passing beautiful windswept beaches and through rainforest between Melbourne and Adelaide - although driving that far will take a few days.
Many people drive for a day from Melbourne, and then come back, taking a night in one of the small hotels along the way, or in a campground.