Hi,
Its hard to specify without some more details
Australia is gorgeous but pretty expensive especially along the coast. Do you have a budget in mind?
Also what kind of weather would you prefer? Cairns is breathtaking and a awesome option but its very hot and humid in summer, your child may not be able to comfortably play outside for long periods in the middle of the day without getting either sunburnt or over heated .
Do you like the coast or the bush? Will you buy a car? Public transport out of the capital cities is pretty hit and miss in Australia, so I would definitely recommend looking into buying a cheap run around.
Holiday letting would be your best bet. There is a huge rental shortage in many parts of Australia and the rental market can be very competitive. With a holiday house you may be able to get a discount for month long stays. Air B and B would be a good place to start your search, or stayz or even just try typing in holiday houses in your preferred location and you should get some hits!
Personally I would look into the Illawarra region of NSW such as Wollongong, Shellharbour and Kiama and the Sunshine Coast in Queensland. The Blue Mountains just outside Sydney is also very beautiful and they should offer lots of accommodation options. All these areas are very, very safe especially compared to U.S standards but of course there may be bad pockets in each town. Just look at crime stats or google maps before agreeing to any lease.
Best of luck!