"Winter". Tee hee...
I'm going to reiterate choosing an area and exploring rather than trying to see everything. A couple of points I would make...
Sydney is normally a very, very expensive destination. But not much business travel happens in January, so accommodation in Sydney itself can actually be a fair bit cheaper (think 'very expensive' instead of 'very, very expensive') than other times of the year. Day tours to the Blue Mountains are pretty easy to sign up for. Alternatively, Leura is a great place to stay for a couple of nights if you want to experience more than a day tour and you can catch the train from Sydney.
Otherwise, catch a ferry to Manly where you can laze at the beach. There are beautiful walks around the foreshore of Sydney Harbour, if that's your thing. In fact, there's an 80km trail from Bondi Beach to Manly Beach (Sydney's two most famous beaches) that is simply stunning. The stretch around the Royal Botanic Garden, past the Opera House, through the Rocks and over the Sydney Harbour Bridge is truly iconic.
Sydney has a few zoos - Taronga is the most famous, but when we used to live in Sydney, we'd take international visitors to Featherdale. At Taronga you'll see animals from all over the globe, but Featherdale is geared towards Australian fauna. It's a bit of a trek, but some Blue Mountains tours will include them on the way. If you go to Brisbane, Lone Pine Koala Sanctuary is another similar venue.
Australians flock to the beach over the summer holidays. Accommodation costs in most beach destinations will be sky high, as Deborah mentioned. North Queensland (Great Barrier Reef area) may not be as booked out as other areas. North Queensland is very hot and humid that time of year and it's stinger season (read up on it, take appropriate precautions) but I would highly recommend seeing the reef if that's where you want to explore.
Unless you are very confident driving long distances, don't. Once you move beyond the Melbourne-Canberra-Sydney-Brisbane corridor, road quality can become quite variable. Most tourists aren't used to driving on the 'correct' side of the road and avoiding kangaroos is as much an art as a science.
I would absolutely avoid Uluru at that time of year. It's stunningly beautiful, but the heat will be deadly. I used to live in the Outback and every year tourists would die from sunstroke, heatstroke and dehydration just by going outside at the wrong time of day, at times within 15 minutes walk of where they were staying. It's hard to describe just how brutal the heat can be.
