What are you doing in Australia for 5 days and what's your budget?

You said you want to see Sydney, Melbourne and Brisbane - you want to do that in 5 days? You do realise that flying between BNE, SYD and MEL - you may as well drop 1 city and spend that money on a decent Reef boat tour.

With the greatest respect, I don't know what you're trying to achieve.

The Great Barrier Reef is not a World Heritage Area for nothing. It's one of the great ecosystems of the world. If you're wanting to see it, I urge you to at least do it properly.

I do apologise if my tone appears a little on the harsh side but perhaps I'm not the kind of traveller who does things "a la Contiki", spending a few hours in 1 city and then moving on and at the end of 3 weeks go "I've seen London, Paris, Brussels, Amsterdam, Cologne, Frankfurt, Stuttgart, Munich, Salzburg, Vienna, Budapest, Zagreb, Venice, Florence, Capri, Naples, Rome, Milan, Zurich and Geneva" and boast about how they've "done Europe" when they've spent 6 hours in each city.

Unless you have specific reasons (like business) for doing SYD, MEL and BNE in 5 days, it pains me to see someone spend 5 days in Australia and not utilise them particularly well.