Many people wrongly believe that they will be taxed on any money they send to Australia, but this is not the case.
You can freely send money to and from Australia without any tax consequences in Australia whatsoever.
Once you begin living in Australia you will likely be subject to worldwide tax, however this will not be on the original amount you had in savings or cash, only on any interest that may have been earned on it after the date you have formally relocated yourself and family to Australia.
Even if you send the money down for family members, it will not be taxable to them as long as it is not payment for some service or employment. Simply gifting them or loaning them money is not a taxable activity.
If you continue to live overseas for some time, then any interest earned in Australia will only be subject to a 10% Withholding Tax.
Once you move to Australia, there is no obligation to close the bank accounts you have in your country of origin, so you can keep them open and leave funds there if you so choose. You will just be required to declare any interest earnings on the accounts each year.
Many people mistake the need to declare cash taken to Australia with some sort of taxation consequence. The Cash Transactions Reporting rules require any one leaving or entering Australia with more than A$10,000 equivalent in cash to declare it at the airport.
This is more for anti terrorism, money laundering and anti crime than for tax evasion.
In fact, you can legally bring in much more cash than the declaration amount, as long as you can show a legitimate source for the money, such as closing a bank account or proof of an official withdrawl.
If the amount is electronically transferred to Australia, rather than sent in cash, then there is no restriction on the amount sent, as anti terrorism measures have already been taken by the sending bank or institution to verify your identity in accordance with international convention.
The Australian tax laws are surprisingly favourable for incoming migrants and you should not be fearful of any potential impact on your savings. This does not mean you can be complacent, and it is advisable to gain a suitable level of understanding of your obligations in order to ensure you legally keep your taxes to a reasonable level.