Australia- Restaurant Recommendations

For Sydney sightseeing:

Visit the Sydney Harbour at night! It is stunning with the illuminated Opera House, Sydney Harbour Bridge, and the city lights.

Walk across the Sydney Harbour Bridge for awesome views of the city and the harbour.

Take a ferry anywhere. We took the ferry to Manly Beach. Had a delicious lunch and walked along the beach front. Best part was the views from the ferry as we were leaving the Harbour. Even better than the views from the Bridge because you get to see the bridge, too, and you sail past the Opera House.

Australian Maritime Museum is outstanding! On any given day you can board and tour 5 ships. Located at Darling harbour.

Art Gallery of NSW is an outstanding art museum that includes amazing art by Australian and aboriginal artists.

Australian Museum is outstanding, too, and includes science, nature, and cultural exhibits especially from various Pacific cultures.

Day trip to the Blue Mountains only if you have the time. We spent 2 nights here which I think is more enjoyable than a daytrip.

Sydney is one of the outstanding cities of the world. Don't shortchange it.

Sydney Restaurants:
Our favorite is The Sanderson. Had an amazing dinner and service is first rate. It's expensive. An anniversary gift from our daughter and SIL.

Emperor's Garden BBQ & Noodles for good Chinese home-cooking.

Nanjing Gourmet Restaurant for food from the Nanjing region of China. Not a gourmet restaurant, though. But very good food.

Melbourne Sightseeing:
Explore Chinatown, Federation Square, and stroll along the Yarra River.

Immigration Museum is very interesting, and I learned a lot about Australia's immigration history.

Queen Victoria Market is the largest open-air market in the southern hemisphere. Has a very pretty newly remodeled food court.

Royal Botanic Gardens are lovely with Victorian era landscaping and a pond.

Hosier Lane is famous for its street art and graffiti.

There is a huge Art Museum in Melbourne. Can't think of the name. We didn't have time to go there.

Melbourne Restaurants:
Cumulus is our favorite restaurant in Melbourne. Everything was delicious.

Higher Ground for a fabulous breakfast. By the way, we noticed poached eggs with avocado are very popular in Australia.

Old Beijing - we had the best Chinese food here.

Piccolo Me is a fast food place that is good for breakfast and inexpensive if your hotel rate doesn't include breakfast.

We didn't go to Hobart and Brisbane.

Ice cream in Hobart: Van Diemens Land Creamery

Among the best I’ve ever had.

The gallery is the National Gallery of Victoria.

On the now quite rare occassions I eat in Melbourne I normally head to the famous pub Young and Jackson's. They do solid pub meals but also have a flasher menu at Chloe's. Chloe is probably the most famous painting outside of the main galleries and is on display at Y and J. Great back story to the painting too.
The pub is on one corner of the Flinders Street/Swanson Street intersection. I believe you had accommodation booked a bit further up Flinders Street so it is well placed for you.
The other 3 corners of that intersection are taken up by St Paul's Cathedral, Federation Square and the iconic "meet me under the clocks" facade of Flinders Street station.
A short walk across the Yarra River on Princes (not Princess) Bridge will bring you to the Arts Precinct including the National Gallery of Victoria.

The Fitzroy Gardens in Melbourne are lovely. I loved the Penguin parade. I agree about seeing the Sydney Opera House at night. Blue Mountains were an ok day trip but I don't think they are a must-see. We loved the zoo in Sydney. If you get advance tickets, be sure to get tickets for the zoo you will visit-there are two Sydney zoos. I think the one we went to is Taranga-it is easily reached by a hop-on hop off boat.

Thanks for all of the recommendations everyone. I’m taking notes.