If you have time..get down to Melbourne. Sydney is a nice city but it just doesn’t have the atmosphere of Mebourne – Sydney is where Australia’s money is made, Melbourne is where the fun is had!
If you do plan on travelling around then get to the Barrier Reef and Ayer’s Rock (Uluru). You can get internal flights to these places (if you’re concerned about your carbon footprint then flying might not be the best option but we’re talking massive distances so I think it’s acceptable). Although driving through Australia is great and you’ll miss quite a lot from the air – if you do fly up to Cairns (to see the barrier reef) try to request a seat on the right-hand side of the plane – when we flew there the plane flew over the reef and it was clearly visible from the right of the plane (I was on the left side and didn’t see anything!).
The Blue Mountains (just outside Sydney) are a closer option – they’re certainly worth it. Take a walk over the Sydney Harbour Bridge (I didn’t get to do it which is a real regret).
Buy some decent guidebooks (my preference is Rough Guide) and take their advice.
Have a good time!
Loved your tip about the internal flights. Thanks!