A paraphrase:

Australians - be happy! Our country is free, unburdened by the past, and our future is ahead of us.
Our land is full of natural riches, and through hard work we can achieve wealth.
This continent has the two-fold benefit of the sea - protection and opportunity.
We have a country full of natural beauty, with animals, plants and landscapes not found anywhere in the wide world.
When the histories are written, let every one show Australia's progression towards a prosperous and fair future.
So Australians, join together and rejoice that your country is building a strong, beautiful, happy nation.

Interesting note: this is the original final verse.

Shou'd foreign foe e'er sight our coast,

Or dare a foot to land,

We'll rouse to arms like sires of yore

To guard our native strand;

Brittania then shall surely know,

Beyond wide ocean's roll,

Her sons in fair Australia's land

Still keep a British soul.

In joyful strains then let us sing

"Advance Australia fair!"

heh