A week allows you to see quite a lot. Here are my recommendations;
1. Spend a day walking around the city itself. Be sure to include the Opera House, the historic Rocks area and Darling Harbour. The view from Centrepoint Tower is also quite spectacular. Tours of the Opera House and the Rocks are both available.
2. Take a cruise on Sydney Harbour - or simply ride the ferry to Manly (almost as famous as Bondi) and back, if you are not interested in commentary.
3. Take a day trip to the Blue Mountains. Either use an organised tour or hire a car. There are many wonderful bush walks to do, if that is what you enjoy, and an overnight stay might then be worthwhile.
4. Catch a bus from Wynyard to Palm Beach, walk around Barrenjoey headland, and return.
5. Don't miss Taronga Park zoo. It is accessible by ferry from the city and its location cannot be beaten for constant glimpses of the harbour. Many animals are now displayed in naturalistic settings and native animals, including the playpus, are well represented.
6. Another wonderful way to fully appreciate Sydney Harbour is to do the walk from Spit Bridge to Manly Beach. It is medium difficulty, and can be accessed by bus from the city.
7. The Art Gallery, Domain, NSW State Library, Parliament House, Hyde Park and Botanical Gardens are all located within the the CBD, and worth a look.
There is no shortage of good restaurants in Sydney, of all varieties. We do have a Little Italy restaurant area in Leichhardt, but you would probably want to catch a taxi from the city (although I imagine you could also catch a bus).
I would not worry too much about driving on the other side of the road, although you do have to be extremely careful in roundabouts and making turns. However, my husband and I have done it quite successfully in the US.
Ask to have bacon cooked crisp, as most Americans seem to find our preference unpalatable (I have heard the word "raw" used).
We have a beautiful city, and no doubt other people will suggest other activities, but I hope you enjoy your stay.