Well, firstly Dec/Jan is summer school holidays here, so be prepared for crowds and book accomodation early! It'll probably be a little expensive too. I'm not sure on accomodation specifically, hopefully someone else will be able to help out with that. You could try looking for a holiday house - http://www.stayz.com.au/
normally has some good, family-friendly places.
Sydney has lots of kid-friendly things - there's the Zoo, Aquarium, a bunch of good museums (the Australian Museum has a couple of fantastic kid-centred exhibitions, including ones on Dinosaurs and scary Australian creatures that are worth a look, and the Powerhouse Museum can be fun as well). But they are both paid admission.
As for national parks, etc - the Blue Mountains have some of the main ones near Sydney, with some lovely walks especially near the Three Sisters (although I'd be careful of doing long walks; summer weather can be brutal for bushwalks, especially round there). Most of my knowledge is based around Sydney (I live there), but there are also some national parks and beaches a little bit north on the Central Coast that I'm really fond of - but I'm a former resident, so I'm biased! The walks in Bouddi National Park near Macmasters Beach are my favourites.
Hope all that helps a little! I can't help with Melbourne, sorry!
Feel free to hit me up with any other questions