If you wan't to study, I'd pick one of the major universities. There are several in most states. If you want to get around regularly and see places, unless you have money to spend on lots of airfairs/plane trips, sticking to the East coast (Melbourne/Sydney/Brisbane) would probably be your best bet. Adelaide would be workable as well depending on what you want to see (it tends to be quieter than the other cities from what I've found but if that's what you're looking for might be worth a look). Again will depend on what you're looking for, for example Tasmania has beautiful wilderness areas, white water rafting, historic areas etc so yeah if that's what you want to see might suit you.
One thing I've found a lot is everyone underestimates by a large margin just how big the place is. Melbourne to Sydney really is a 10-12 hour drive (if you drive straight there with no detours) and that's probably not even 1/3 way up the side of Australia and traveling on decent roads, so how much time you have for traveling and how much money for things like airfairs/petrol etc can be important.
Also consider the climate you want, as silicon was saying North is lovely in Winter, and stays hot in summer but also gets very humid at times. South can be a cold and wet in winter (but it's not too bad, rarely snows at sea level) and gets plenty hot in summer.
Have a look though the university sites. Many have an international student's page that will tell you what to expect about the city and student life in general. Different universities have reputations for being better at certain things than others, and also having different approaches to teaching, so knowing what you want to study then asking around may also help. Sorry most familiar with the Victorian Universities so I'm sure I've missed a number of them, but some larger ones include The University of Melbourne, La Trobe University, Monash University, Deakin University, RMIT, Swinburne University (Victoria), University of Sydney, University of Queensland. (I'm sure I've missed a number in the other states but that's a start)
It should be fairly easy to get around, although word of warning, the public transport system is probably not nearly as good as what you're used to from what I've been told, however it's workable, and if in doubt ask, people are generally pretty approachable as a rule. *laughs* as Celia says probably ask a second oppinion on that, I'm probably biast since I live here.
Australia is and isn't like the US. I haven't been to the US so I'm going off what I've been told and know of the two places. Have to be a bit more specific as to what you mean. We have a number of similar things such as food places, TV shows, clothes brands etc. There are similarities but also some distinct differences in the culture as a whole. For example I've been told our sense of humour can sometimes be a bit wierd, there's some odd slang and I'm sure a few other things nothing too drastic.
Anyway hope some of that helps
