You most likely will not be eligible to apply for a student visa until after your application has been accepted by the host institution and/or you receive relevant visa documents. However, you can start researching the visa requirements earlier by viewing the country's embassy materials regarding visas. Take particular note of how long the application process will take,  whether you will need to book an interview (and how long the wait time might be), and preparing all required documentation ahead of time.
For example, it is important to note that some countries have consulates/embassies only in Canberra. International students going on exchange may be required to provide additional documentation to their host university, such as having their CoE notarised then getting an Apostille through DFAT.

In some cases, you will not be able to apply for a student visa or residence permit until you have arrived in the country.

Please make sure you have a passport with at least 6 months of validity remaining after your planned return date to Australia. Passport renewals can take several months so ensure that you will receive your new passport in plenty of time to apply for your visa.

If you have dual citizenship, be aware that this may cause difficulties in receiving consular assistance if you choose to travel on a non-Australian passport. Read this information about dual citizenship.