G’day,
Great answers from the first answerer describing UWA and Uni of Sydney. Now for a more un-biased info… 
There are only 39 universities in Australia and they are very tightly controlled, therefore the quality and recognition of their graduates are equal from wherever university you are studying from. This means that there is no BEST university in Australia, as they are comparatively equal. The most important thing is you have to READ the course information carefully, since some courses may have the same name but different content. You may also want to choose the city that you want to go as well.
Living cost… Sydney and Canberra have the highest living cost, followed by Melbourne & Brisbane, then Perth, Darwin, Adelaide & Tasmania the lowest. I am attaching a weblink from one of the uni in Melbourne as an indication.
To find out the course that you are taking and where it is offered, I suggest you to go to IDP Education Australia website (www.idp.edu.au). IDP is an organisation that gives information to international students who wants to continue their study in Australia. Once decided on the uni, fill in the application form and send it together with certified copy of your academic qualification (in English and original language).
Hope this helps. Good luck for your study and welcome to Australia 
I’m currently living in Perth, and attending the University of Western Australia – known for being one of the best Universities in the country. It offers such a wide range of courses, accomodation and extra activities concerning sports/ theatre/ spirtuality/ food/ night life/ entertainment/ ect… for people of all different nationalities and ages. The list just goes on of what UWA has to offer. The campus is also large and beautiful – housing gardens, lawns, sports fields, and native Australian plant-life and animals (including a ton of very friendly peacocks!). I’m at UWA to be near my family who live just outside of Perth, but whilst looking for a University, I also investigated Sydney. My personal favourite was the University of New South Wales, as there was such great variety of courses they had to offer (I was looking into studying Media & Communications), but if I were to take courses geared more towards the Sciences, The University of Sydney would of been my choice. Probably the most prestigious University in Australia, when entering, it feels as though you have stepped onto the set of Harry Potter. The campus is certainly picturesque and is in a great location according to the city. However, it is extremely hard to get into – you must have excellent marks.
References :
G’day,
Great answers from the first answerer describing UWA and Uni of Sydney. Now for a more un-biased info…
There are only 39 universities in Australia and they are very tightly controlled, therefore the quality and recognition of their graduates are equal from wherever university you are studying from. This means that there is no BEST university in Australia, as they are comparatively equal. The most important thing is you have to READ the course information carefully, since some courses may have the same name but different content. You may also want to choose the city that you want to go as well.
Living cost… Sydney and Canberra have the highest living cost, followed by Melbourne & Brisbane, then Perth, Darwin, Adelaide & Tasmania the lowest. I am attaching a weblink from one of the uni in Melbourne as an indication.
To find out the course that you are taking and where it is offered, I suggest you to go to IDP Education Australia website (www.idp.edu.au). IDP is an organisation that gives information to international students who wants to continue their study in Australia. Once decided on the uni, fill in the application form and send it together with certified copy of your academic qualification (in English and original language).
Hope this helps. Good luck for your study and welcome to Australia
References :
http://www.deakin.edu.au/international/prospective/coming_australia/money_matters.php