The best university in Australia is the University of Melbourne.
To dispel some rumours in this thread Melbourne does have beaches and hot girls. In fact one of our biggest tourist attractions are our beaches.
Canberra has ANU which is a great university, it has somethings to do but if you're not going to be in Australia for at least a year than it would be better to go to a bigger more vibrant city such as Melbourne, Sydney, Brisbane.
Perth is a really pretty town but there isn't much to do over that side of Australia. Although you can see the Great Barrier reef and some cool animals. If you want a party lifestyle, city living, urban experience than Sydney or Melbourne would be best. Because they're big cities there are a lot to do and they have a lot of things angled towards tourists that are easily accessible.
The best public transport in Australia are in Queensland and Victoria (Brisbane and Melbourne), Sydney's public transport system is atrocious. So if you're not going to get a car one of those cities would be best.
Melbourne is also a lot cheaper to live in than Sydney if that weighs into your decision. Both cities have their individual merits. Melbourne is more laid back, Sydney has an intense vibe to it everywhere you go. The night life is also a lot better in Melbourne than Sydney.
As someone noted the snow is closest to Canberra then Melbourne (2 hours and 2hours respectively). You can get buses to both of the closest mountains from these cities.
Personally I wouldn't go to Canberra, it is too cold and there isn't enough for me to do. But it does have good academics and I've heard they have some great parties.
I've made this decision myself and decided to go to school in Melbourne even after getting higher scholarships to Usyd and ANU.
If you have any specific questions feel free to hit me up.