The best time to visit Australia is during the southern hemisphere summer months (November to March).
Australia is a well developed country and has a large number of tourists visiting each year. Unless you go to a small town you will easily find lots of options for hotels and restaurants. There is a huge variety of high quality locally produced food options. Unless you are staying on a sheep farm you wont have to eat lamb or mutton.
As others have mentioned there are restricted shopping hours once you get outside the main tourist areas. Even in big cities like Brisbane and Perth it can seem pretty quiet on the weekends. If shopping is your priority then maybe Hong Kong is a better option.
The main places that people visit and unique attractions in Australia are:
Canberra - is the political capital of Australia and home to national institutions like the Federal Parliament, War Memorial, National Library, High Court etc. It is built around a large artificial lake (Lake Burley Griffin).
Central Australia - This is where Uluru (formerly known as Ayers Rock) is situated. There are also other desert formations and National Parks.
Melbourne - is the sporting capital of Australia and home to many different sporting events like the Australian Open tennis, Melbourne Cup horse race and Melbourne Grand Prix. If you visit between April and September you should go to an Australian Football match at the MCG.
North Queensland - this is where the Barrier Reef is located which is great for Scuba diving. There are also rainforests, beaches and island resorts. It has a tropical climate.
Sydney - biggest city, has a beautiful harbour which you can cruise on, the Opera House and Sydney Harbour Bridge. It also has an Aquarium/wildlife park in the city center and another Zoo where you can take a behind the scenes tour and get close to native animals (kangaroos, koalas etc). It has many beaches, restaurants and night clubs.
Western Australia - This is the largest state, the main city is Perth but if you go outside the city you will find wonderful places to stay like the Margaret River wine district (the best in Australia IMNSHO) in the south and Monkey Mia beach in the north (famous for friendly dophins). You can take a ferry from Perth to Rottnest island which has unique local marsupials (Quokkas).
It really depends on how much time you wish to spend here and the types of things you like to do. There are lots of other places I haven't mentioned but they are not particularly unique in terms of their attractions.
If you are only staying for a short time Sydney is the best place to visit. You can get to Canberra from Sydney in a 1 hour flight or 3 hours by bus/car. There are also wineries, national parks and other attractions within 100 mile radius of Sydney. All the other places I have mentioned are quite a long way apart.