I chose to go on exchange because it's a great opportunity to travel and to experience what it is like to live for a longer period in another country. Australia felt as an exciting choice and the University of New South Wales is a good and reputable school.
I began to search research teams when the spring semester ended in June so that I had plenty of time contact as many supervisors as possible and decide which one I would choose. In the end, I chose one of the first groups I contacted, so there was really no hurry. I know many how contacted their supervisors in the last minute and still got an internship, so don't panic if you are bit late. Anyway, it's always better to be out in time to be on the safe side. I also recommend you not to settle for a group just because they where the first to reply or you don't feel like you have the effort to ask several groups. Instead, take contact with as many as you can (that seem interesting to you and try to find out as much about the projects they offer as possible. Since you are going to be there for six months, more or less, it is important that you enjoy what you are doing and the people you work with. It is a good idea to ask a bit about the group and maybe choose one that have students or young PhDs since you are going to spend a lot of time in the lab this will make the whole exchange a lot more fun.
Once you have found a group that accepts you, you have to apply to the school as well. This is just to get you signed in as a student and for you to get a student number that you later on can apply for your visa with, since you more or less already got accepted to the school trough the nomination from KI. This online application was quickly done and easy to understand. Things to consider when filling it in is that the passport number must be the same as the one you apply for a visa with. Therefore, make sure that your passport is valid for the entire period you intend to stay in Australia.
After you have received the acceptance letter from the school, you can begin to apply for student visa. You usually receive your visa within a couple of days after your application (I got my visa for less than 12 hours), but they recommend that you apply at least a month before departure. This is because they do random 'extra' controls which can take up to 4 weeks to complete . There are lots of different visas to choose from but I would recommend a student visa because it is cheapest (costs around 1700). I applied for student visa subclass 507.
Something that is important to be in good time with is flight tickets, as they often rises quite dramatically in price. I would also recommend to buy tickets from one of the more exclusive airlines as there isn't that big difference in price and it's really worth the money for the comfort when you fly for 20 hours. I flew with Emirates and was really satisfied. They had comfortable chairs, "good" airline food, and lots of new movies / series you could choose from on your private TV screen.