You can register a trademark in these three countries in three ways:
File three separate applications
This will require some legal assistance. Both USA and Australia require non-citizens to have a local representative. The EU doesn't have this condition, but applicants must submit a correspondence address within the European Union (EU) or the European Economic Area (EEA). Depending on where you are from, you can seek local attorneys that will handle your case.
Find a registration provider that has a license in all these countries
This is basically option 1, with the difference being that you don't have to search for local attorneys; you can find a law firm that can handle all three of your applications. Besides convenience, you can sometimes get a multi-country discount.
Apply through WIPO
You can use WIPO's Madrid system to apply in all these countries simultaneously. However, you'll have to base your application on an existing application in one of WIPO's member countries. In practice, this means you can file an application in one of these three countries and use the system to apply in the other two. However, this is not financially optimal as WIPO tends to be a better option if you apply in five or more countries.