Hello travelsawait101!

My response to your question would include:

• Arrive at the airport in plenty of time and as far as possible give yourself plenty of time at connecting airports. (There are some frequent travellers who seem to regard it as a badge of honour to arrive as late as possible at the airport, but check-in and security can be frenetic enough without having to rush and worry unduly. So what if you have to wait around at the airport for a while - you'd just be waiting around at home otherwise.)
• Think about what you might want to do during the flight. Most long-haul flights now generally have at least reasonable in-flight entertainment, but I like to have my iPod (with noise-cancelling headphones!) with me, and a Kindle (multiple books in a very light container, so I can choose what book I want to read as the fancy takes me).
• Aim to eat properly but not too heavily both before and during the flight, drink plenty of fluids (preferably water) and go easy on both alcohol and caffeine-containing drinks. Eating well and limiting alcohol will make you feel better, and tea and (especially) coffee tend to promote anxiety anyway if you're already in a situation where you're prone to be anxious.
• I try to have the things I know I'm going to want during the flight packed in a small bag inside my carry-on bag that I can take out before I sit down. (I'm thinking of things like book/Kindle, iPod etc, spectacles case, any medicaments you think you might need - not much, but it saves you having to get up and down and hunt for the things in your carry-on bag.)
• I know it's a truism and not really the focus of any anxiety about flying, but remember that the most dangerous parts of the journey are getting from home to the departure airport and from the arrival airport to wherever you're going!
• Have a pen handy - you'll need one to fill in the landing form before arrival into Australia.
• And, perversely perhaps, try to think of the flight as a chance to sit and relax for several hours, something that we probably don't do enough of in our everyday lives. You will almost certainly have been busy in the lead-up to the trip, so the chance to relax might be doubly welcome. Admittedly this mind-set might go only so far on a trip as long as the one to Australia, but it is a good state of mind to start with and can set the tenor for the whole flight.

I hope the flights (and the visit to Australia) go well.