Hi Valerie, there are plenty of free things to do in any city and Sydney is no different. 

You can ride the ferries and trains to get to most destinations. Buy a travel card from any news agency or ask where to get one. 

Walk across the Sydney Harbour its free and the views are fantastic.

Ride the elevator to the bar at the top of the Shangri-La Hotel, Sydney. Drinks are expensive but the view is free. 

Take a walk along Manly Beach much nicer than Bondi. 

Not sure about the cost of museums or art galleries. Send me further questions if you want more ideas. 

Riding the ferries is a great, inexpensive way to see the city and get a feel for Sydney life. Manly is a great ferry destination and a great beach town. I personally love the @Sydney botanic gardens which is a great place for a stroll, picnic, etc. you can walk around The Rocks (touristy part of Sydney but interesting nonetheless), to Circular Quay (the main ferry terminal), then from Circular Quay,past the Sydney Opera House to the botanic garden. Easy, cheap and lots of sights.

Try to book yourself a free greet with the Sydney Greeters. Part of the Global Greeter network, these volunteers spend a couple of hours showing your their favourite parts of the city.  It's a really good way to see something a little different.   
Also for a cheaper drink with a view visit The Glenmore Hotel after your walk across the bridge.  Then visit Foundation Park and The Rocks Discovery Museum 

I made a self-guided walk you can follow if you have a couple hours to spare

Depending on what time of the month you arrive, you may be able to catch VIVID. 

Like people said below, all the beaches are free. 

QVB shopping center might be worth checking out, if you like architectural stuff. It's just a fancy shopping center by Town Hall Station.

The Bondi to Coogee Walk is by far the best free thing to do. Take the bus to either end and walk down the coast. The Royal Botanic Gardens Sydney is another great sight with views of the harbor. The Art Gallery of Nsw and Museum of Contemporary Art are also free, apart from special exhibits. There's a free tour daily from Town Hall. Here's some more free and cheap things: http://www.carolineinthecityblog.com/2013/05/08/best-free-and-cheap-things-to-do-in-sydney/.

Hi Valerie,

Here are some suggestions:

1. As others have mentioned, take a ferry to Manly, or to Watsons Bay. Not free but pretty cheap.

2. Walk across Sydney Harbour. It is a spectacular walk.

3. Take a free tour of Sydney. E.g. www.imfree.com.au/sydney/

4. For outdoor activities, take a look at: Http://www.sydneyoutdooradventures.com

5. Art Gallery of Nsw - many parts of the gallery are free.

I hope you have a great trip!