There is no need to always spend money on entertainment. Sydney have many places that are great and you can visit them for free. Check those museums and art galleries that are definitely worth visiting!
The White Rabbit Gallery, Chippendale
My favorite gallery in Sydney. This place is created by Judith Nelson, who was traveling to Asia and felt in love with artwork. She use the former depot for Rolls Royce to establish a contemporary art gallery with pieces made by Chinese artists. Have to point out that every 6 month the exhibition changes, so I like to visit back here often.
MCA – Museum of Contemporary Art, Sydney Harbour
It is a great stop while you are in Sydney Harbour. Gallery has permanent exhibitions as well as temporary displays. In MCA you can learn about art of indigenous people of Australia and artist from all over the world. There are regular workshops and organised event for children – you can find information about the events here.
Once you are here remember to go on the top level – there is a rooftop café with view at Opera House and Harbour Bridge. On your way out, I recommend you to stop at the gallery shop. You will find beautiful photo albums, educational toys and things that will make a great souvenir or gift from Sydney.
National Maritime Museum, Darling Harbour
The museum is located in beautiful Darling Harbour, perfect for a walk. In the main building of the National Maritime Museum you will discover exhibits and stories about ships and sailing. For example, you can see the sailing boat of the first woman who sailed the world by herself, Kay Cottee. There are few levels of display, and also big ships on the harbour to explore (although for that you need to purchase a ticket).
You might also want to climb on the lighthouse in exchange of ‘golden coin’ donation. From there are nice view on Sydney Harbour.
Sydney Observatory, Millers Point
This is the heritage listed place with a lot of history to learn. Back in time the observatory was used in shipping, navigation and study about starts. The Observatory Museum organize tour options and late night event like star gazing (this requires booking and fee).
Observatory Hill is a beautiful spot to sit and look at the Harbour Bridge and the bay. A lot of people comes here for picnics or photo shoots.
Carriageworks, Redfern
Industrial area where you can enjoy contemporary art exhibitions. The building is very spacious, mostly of the time you may see big art installations in Carriageworks. In addition, there is Farmers Market regularly on Saturdays and occasionally other events (like night market, etc.).
Australian Museum, Darlinghurst
The museum is located near Hyde Park, it’s a nice place too. Australian Museum is big space, where you can learn about the Australian history. There is Aboriginal art and knowledge about Australia`s first people, recent history about colonization and treasures of most influential people in the country. There is a gallery that shows stuffed animals and gives a lot of interesting information about them (most of the them were donated to the museum from the zoo, after passing away).
Moreover, there are temporary exhibitions like photography contests, history or exhibits of dinosaur’s. Some of these temporary exhibitions might require paid ticket.