Australia and especially Sydney is expensive! Already before departure I spent money on the visa (535AUD) and flight tickets. As mentioned, I did not need any vaccinations or medical examinations which saved me a lot of money as I heard from other students that they are expensive.
The main cost is Sydney is accommodation (see accommodation). If you rent an on-campus room, it could cost you as much as 400AUD a week(!) and even a shared room might cost 150-200AUD depending on the neighborhood. My single room in a neighborhood very close to UNSW cost 230AUD a week. The only option for a slightly cheaper room is living in a cheap neighborhood. However, not every cheap neighborhood in Sydney is safe.
Another cost could be public transport. As an international student you are not eligible for the student tickets and also do not try. The fines when you are caught are about 200AUD (Aussies love there fines and you could get a fine for anything). To keep the public transport costs down, I bought a secondhand bike. In addition, the public transport system in Sydney is quite bad. Cycling to the city was faster than by public transport. Sydney does not have a metrosystem and only some neighborhoods are serviced by trains like the pendeltag. All other neighborhoods including UNSW are only serviced by busses which never run on schedule. Traveling from the outskirts of Sydney to UNSW could take up to 2 hours one-way. So make friends with Aussies that have a car and can drive you anywhere :). On topic again, as I did not use public transport daily I always bought traveltens, which are ten one way bustickets you can buy anywhere (but in the bus). Trains are relatively cheaper than buses. There are special weekly/monthly tickets for foreign students as you are not allowed to use student tickets, but you should ask the international coordinator for more details.
http://www.gumtree.com.au/ is a website you should have in your favorites: you can find anything here: from secondhand bikes and furniture to rooms for rent. And eventhough the traffic is sometimes crazy and the hills often high, I used the bike a lot. One of the main reasons was to do my groceries. Coles and Woolworths are the main supermarkets in Australia (like Ica and Coop), but again, they are quite expensive. Instead, I often cycled to Aldi, which is a German store very similar to Lidl and also cheap. Next to 'my' Aldi was a cheap vegetable store and the veggies here were a lot cheaper than in the supermarket.
The student Union, Arc, is free to join and their membership gives discount on various things. Another expense many foreign students complain about is alcohol. The price of alcohol is as expensive as in Sweden, but the bar of the student Union is a lot cheaper, especially during happy hour.
Cheap flight tickets can be bought at http://www.tigerair.com/au/en/ . I never flew with them, but apparently they are like Ryanair but slightly better. Another cheaper company is http://www.jetstar.com/au/en/home that fly to more destinations in Australia and New Zealand.
If you study in Australia on a student visa, you are allowed to work. I decided not to work as I made very, very long weeks at the lab and wanted to use my free time to explore Sydney and surrounds and have a social life too. Apparently, it is not easy to find a job as exchange students have to compete with the many backpackers in Sydney. However, this is what I heard and not my own experience.
When selected by KI, you will receive a scholarship. This scholarship entails more than just the sum of money they mention. You will also get your application fee of 330AUD which you have to pay when you apply for your student card refunded. In addition, you receive an Insurance and because of this Insurance you do not need to buy an OSHC which is an additional 250AUD (approximately).
Overall, the cost of living in Sydney is higher than in Sweden, mostly because of the very expensive accommodation. It is not usual for project students to receive any financial compensation from the lab. So make sure you have some savings or are able to get extra money from any source (student aid from your home country, scholarships, (grant)parents, ...).