Travel tips?

[speedeight]

G'day (sorry!)

I'm going to be in Australia from the 27th Dec to 19th Jan and I'm after some travel advice.

We'll be in Sydney until the 3rd Jan, but after that want to travel north up the east coast towards Cairns. Is this too much to do in 15 days?

We're prepared for internal flights if necessary but would like to do the backpacking thing properly as much as possible.

The things we definitely want to do is visit Brisbane, the barrier reef, Australia Zoo, Fraser Island. Would it be a good idea to fly from Sydney to Brisbane and then make our way up to Cairns from there? That would save time but would we miss anything we really shouldn't?

Thanks for any advice!