Mind the Wallabys Mate, they're a dangerous breed.
Dont' try to play the digeridoo, and what ever you do, don't feed the Kangaroos - mean suckers they are.
Other than that you will do fine.
The end of this month me landlord is Going Down under and every day this week I' been talking Australian at 'im, "Throw a Shrimp on the bar-be" and such phrases that the Natives speak Down Under, told 'im I' keep an eye on 'is dingo* too.

As for Depression and anxienty, they both go hand in hand two facets of the same gem. Take deep breaths, count to ten slowly and you will do just fine.
Trips are stressful no matter the distance. I suggest making lists, like lists of things to pack (an inventory if you will) a list of things you want to see, a list of your hotels/car rental, airline tickets, passport, traveler's checks, etc with addresses (for pass port and traveler's check, local near your hotel) and phone numbers. Carry that in a "safe" place, other than your wallet or purse.
Thus you are "prepared" for the worse so you can "do something" - this impowers you, this provides you with the knowledge that you have things under control - it all helps to relief travel anxieties.
* His dog - I always watch the dog when he travels, like 6 times a year, abroad, cruises back east.... No anxiety for him
