Every time I have international guests over to visit I immediatly take them out to my home range in the Blue Mountains and show them the least touristy places there.
If you want to do it all yourself, Katoomba is only 2 - 3 hours out of Sydney on the train and costs about $AUD20. Thats where ALL the tourists go. If you had a car you would be able to explore the more amazing parts yourself that most people don't see. Take the Grand Canyon walk from Blacheath, organize a Canyon tour with a difference (Bushwalking through a gourge in a wetsuit!!
You'll love it I promise!) See the Hartley Vale and have lunch in the Comet Inn, Jenolan Caves is a bit further out 3 - 4 hours from Sydney but well worth the trip!
Either way get yourself to the mountains. It will be cooler too, cause it's going to be a HOT summer this year!!
Newcastle is only 2 - 3 hours by car or train from Sydney. It is renowned mostly for its industry (Coal & Steel) but these have slowly faded away leaving just some beautiful beaches and great food and pubs (Beaumont st especially!!) Take a trip up to Nelsons Bay while your there to get away from it all and maybe have a skinny dip on a remote beach
Australia Day in Sydney - No drinking on the streets and if you want to see the crackers, get your spot EARLY!!!
Sometimes the more obscure places are the best places to see them from - Try Vaucluse or Balmain. I've always wanted to do a boat tour around the harbour but haven't yet.. I'd imagine that would be the best place.
In two weeks you will be lucky to get out of NSW/Sydney Area without flying around the country at extorsional cost and wasting a lot of time in and around airports. Stay in the Sydney area, there is a LOT more to do in two weeks here!
I'm a local and have lived in Blue Mountains, Newcastle and currently Sydney. I have also travelled the East coast quite a bit so I'd be happy to answer any further questions you have or al least try and point you in the right direction.
Have fun!!