If you are planning on checking out the Great Barrier Reef, I highly recommend you do this from Cairns. The reef can be accessed from other places in QLD such as Townsville, Mackay, Rockhampton, etc. However, having previously lived in North Queensland (and in a rival city: Townsville), I feel Cairns offers the best value as you not only can you can visit the reef, but you can also check out the Daintree Rainforest, Atherton/Tabllands. It's quite amazing as you have the reef on one side, and a beautiful rainforest on other. You can take a scenic train up through rainforest up to a small town called Kuranda, and then come back to Cairns via Cable Cars, or vice versa (this is a day trip). In the cable cars, there are stations which you can stop at on the way, and at these places you can learn about forest, walk around and check out scenry and wildlife. Makes for a great day out!
Other things do include visiting Tjapukai Aboriginal Culture Park if you are interested in learning about the Australian Indigenous Culture. One of the cool things is learning how to throw a boomerang. You can tour rain forest in army ducks, hot air balloon rides, visit Port Douglas, etc.
One last thing, Cairns does have an international and domestic airport; going back to Sydney will be no problem. If you only have one day to spend in North QLD, in my opinion, it doesn't matter too much as to whether you go to Townsville or Cairns. Snorkelling can be arranged, if not you can take a tour on a glass bottom boat or semi-sumbarine (this is a lot better!)
https://plus.google.com/+SkyrailRainforestCableway/photos?hl=en (Sky Rail, I don't think the pictures do it justice).
If you want more info, feel free to PM. I live in Brisbane (QLD) now, which is in South Queensland, and can offer advice if you are planning to visit this region as well.
Thanks for the info. My son actually mentioned learning to throw a boomerang and I mentioned that he should learn to play and didgeridoo and learn about the aborigine. I will need to study a map and try to plan according to region given the limited time. I think we will mostly suggest what we would like to do and it will be planned out for us. It could be very interesting as it may be all planned out for us and we just have to show up. Thanks again.