Are there any limitations to giving gifts of money to our children? And what are the legalities involved in doing so? We reside in Australia

If you are a U.S. citizen or resident, then yes, the U.S. has gift tax.
Read this:
Your U.S. Tax Return: The U.S. Gift Tax

This assumes you are U.S. citizens residing in Australia.

You can transfer up to (I believe) $12,000 each year with no tax consequences or reporting requirement.

That is $12,000 per parent to each child, so a couple could give $24,000 to each child. If you get the grandparents involved, it could be as much as $72,000 per child.

If you exceed this limit. You have to file a gift tax return, but you can avoid PAYING a gift tax if you use some of the Unified Credit. This is the credit that your estate uses to exempt the first $2 MILLION of the state from estate taxes.

If you are Australian citizens, you need to get tax advice from a local Australian tax professional.