Great Barrier Reef
 

We would like to go to Melbourne, Sydney and Great Barrier Reef....suggestions on how to navigate this?

Fly!

The perfect one word answer, Margo_oz.
Australia is BIG and Melbourne is at the equivalent position of about New Orleans and Cairns is up past the Great Lakes in Canada.
Anyway we did a Quiksilver tour out of Port Douglas to the Agincourt Reef which was spectacular overall. Snorkelling was good but I would recommend being a reasonable swimmer. You are out in the open ocean in a roped off area and they take safety very seriously but I was a little unnerved by the huge ocean. I really enjoyed it but no way would I take any risks by swimming too far away.
The helicopter tour is awesome. Wait until later in the day and you might get it for half price.

Come in the Australian winter.

do some research about what else you want to do. See Australia.com

You should directly come to Queensland's central coast where you can visit Whitsunday Islands form a part of the Great Barrier Reef.