I gave up my Canadian residency a few years ago. Along with speaking to your accountant about what you need to do, I highly recommend speaking directly to the Australian gov and tax department to ensure you fill out all the proper paperwork and that you are very clear as to what you and your family will be forfeiting and giving up, and also what you are still entitled to. Since your wife and children are your dependants, they will most probably have to give up their residency as well-- and therefore you all need to be clear and on the same page about doing this.
For myself giving up Canadian residency was somewhat of a simple process of doing all the necessary paperwork, plus shutting down a business I had in Canada, closing canadian bank accounts, discontinuing subscription to medicare (canadian socialized medicine), proving I have no assets, no possessions, no dependant family, no real estate, etc.. nothing remaining in Canada. I did not have to show residency or a visa of another country in order to terminate my Canadian residency.
You should also inquire about how to regain your Australian residency if you and your family decide one day to move back-- find out how long it takes, what you need to do, will you be entitled to Australian socialized medicine right away and if not, how long you have to wait, etc..
Also keep in mind that if you are no longer an Australian resident, you and your wife might have to give up your Australian driver's licenses (you might be able to keep it, but you probably won't be able to renew it)- that is something to look into if you need a drivers license and would you be able to get one from another country without being a resident, or could you get an International driver's license from Australia without being resident, etc... It's little details like this that need to be thought through and investigated-- so important to get a full list from gov. of what you will give up & not be entitled to and make sure you can get these things in another country.. Same goes with bank accounts and credit cards, some countries have rules that in order to terminate your residency you have to close all bank accounts and credit cards as a means of proving you are leaving-- will you have to do this and where will you open a bank account and do you need to have residency in the country you are moving to in order to do so...? Do think all these things through and make a big list of questions for your accountant and Australian government!!!!!!!