Yes it can be done,but with difficulty.We were lucky to be staying with friends so saw more of the places than we would have done on our own.
Drive to Watson Bay for fantastic views and a goanna walking over your feet.Use the local transport.We stayed about 45 mins out of town so got a daily travel pass for $15 which was good for trains buses and ferries.The ferries are a great way to get around,it's a massive harbour with lots of places/beaches to visit.
Take the Harbour Tour with Captain Cook,excellent value.Sit on the Quay to people watch and be entertained by buskers.We saw the Queen !!!
Visit the Aquarium,not usually my scene but set out and managed beautifully.
Have a "lift" Tour at the Opera House.If you have any walking probs,just ask at the desk,same price.There were 4 in our group and we just popped in and out of lifts to get to next level..easy on the legs and we got to see a lot more backstage areas.The normal tour consisted of about 25 people,rushing to keep up with the guide as she climbed the 200 steps inside and straining to hear what she was saying...no contest.
See Sydney at night,especially around the Harbour.
Have coffee and cake at the Intercontinental. We arrived to a barrage of FBI /police/helicopter/activity, How were we to know Ms Rice was staying upstairs??
Spend the day in the Blue Mountains,you need a sunny day though to see the "blue".
Scare yourself to death with a ride on the Mountain Devil at Katoomba.
Lots more to do and see but enough for my poor foot.I did see a podiatrist re foot,(worth every cent) it puts our Health service to shame.

Don't forget to check out Sydney's Underground

For Shopping, the lower level of Queen Victoria Building (QVB) houses Victoria's Basement, a great bargain hunter's heaven.

From the lower levels of Sydney's train stations you can find a maze of corridors that house shopping, wonderful eateries, food shops, flower stands and services like laundry and dry cleaning, wine shops etc. Go explore!

My family and I recently returned from a trip to Sydney, Australia. When I mention Australia to friends the first thing they say is what a long flight. This is the truth, coming from New York we were flying for about 24 hours. Once we landed in Sydney, I quickly discovered it was well worth the long flight. With only a few days to enjoy the city before embarking on a cruise to Auckland, New Zealand my family of six were extremely busy.

We stayed in a division of the Southern Cross Hotel Chain. It was perfect for a six-person family as we had an apartment with a gorgeous view of Chinatown and Darling Harbor (Two of the many must see destinations in Sydney).

The best way to determine where to visit in Sydney is to examine the entire city at once. Impossible you say, not when you can climb the infamous Sydney Harbor Bridge. Through the catwalks and stairs my whole family climbed up and back pausing for gorgeous views of the entire city of Sydney along the way. This is a great family activity for families where all members are above the minimum age of 13. My mom was incredibly scared, but the safety is unlike anything I have ever seen. We were harnessed in and it would have been impossible to fall. They even had a practice run on a catwalk 5 feet above the ground and a Breathalyzer test was mandatory. The view of Sydney Harbor, the Sydney skyline and the Sydney Opera is unparallel, especially at dusk with the sun setting in the background. Dusk or dawn is the optimal time to climb the bridge, the tours are offered 24 hours a day with lights attached to your helmet for night tours. It is by far the most amazing and number one attraction in all of Australia.

I mention the Opera House, another must see, mainly from the outside, the inside is well decorated, but nothing amazing. The Sydney Opera House is known for its parabola or sail design as it rises over the very busy Sydney Harbor. The outside is covered with two different shades of white tiles in order to appear glamorous at all times of the day with sunlight from all directions. Besides up close, the best place to take pictures of the Opera House is from the ferries that run through Sydney Harbor (see pictures). Ferries run from Circular Quay to the popular destinations of Bondi and Manly beach, and the Taronga Zoo.

Speaking of beaches, Manly and Bondi Beaches are the two most gorgeous beaches in the South Pacific. The water is crystal clear with beautiful waves. And although no one in my family is a surfer, it was clearly the activity of choice when we were there. Certainly worth the ferry ride and scenic trip.

While you are at Circular Quay going to the beaches, the other destination accessible by ferry is the Taronga Zoo. I highly recommend the private tour where they show you behind the scenes of all the native Australian animals. We were able to pet wombats, koalas and kangaroos as well as soak up information from the knowledgeable tour guides. A tour is really the only way to properly experience the Taronga Zoo.

Another important stop is the Sydney Aquarium in Darling Harbor where you can go under the tanks of water to see all the marine life. You can even find Nemo and all his friend’s as they are all Australian fish. (There is a special Finding Nemo exhibit at the aquarium that was thoroughly enjoyed by all members of the family).

So now that the touring is over, it is time to shop and eat. We found great shopping at the Queen Victoria Building, essentially a mall. They also had great food, but by far the best food can be found around Darling Harbor. All the restaurants have outdoor seating, which is optimal for the summer weather they were experiencing. For a quick bite to eat during the day, a stop at Harry’s Carte de Wheels is essential for the best potpies in the world. All in all, the food was great, be adventurous, I tried kangaroo and alligator for the first time and they were delicious. Go for it, your down under.

Other famous attractions include the Botanical Gardens, Chinatown, the Sydney Olympic Park and local nightclubs. With so much to see and do in Sydney much more time was needed than the few days we allotted. Plan to spend at least a week or two in Sydney to truly enjoy the city for all she is worth.

The best way to quantify our trip to Sydney, Australia is that when I look back there is nothing that anyone in my family would change. It is the perfect city for people of all ages. And above all, the people are some of the nicest in the world. I can’t think of a better place to vacation than Sydney, Australia.

We, my youngest son and myself visited Sydney last year and found Sydney a very clean, thriving city (and Australia). We were impressed by the punctual public train, bus and ferry service (so was Qantas). We moved about not like a tourist, more like a Sydneysider. The places we stayed in were reasonable priced, very clean with a friendly service.
We have not found anything to criticise in Australia (we weren't looking for either), but can recommend it only in the highest terms. In three weeks I took some 3525 photos!!! If you go there as a guest, behave like a guest and you will find it to be one of the finest countries in this world with the nicest people. I found some "mates".