Consider your reasons. Study abroad programs to Australia and New Zealand will want to know your reasons for wanting to study in these countries. Speaking English and wanting warm weather will not suffice. Try to find a particular area of Australian culture that interests you, whether it is political, literary, or cultural, that you can focus on later in your application essay. If nothing jumps out at you, start looking at particular programs and professors at different universities. The opportunity to work with one of them can be a compelling reason to study abroad as well.
Research programs. The best place to start researching potential study abroad programs in Australia and New Zealand is to talk to your school's office of international programs. The staff there should be able to inform you about your school's programs in these countries (which will most likely make it easier for you to get into) or if there is a program run by another organization that students from your college have done before. In addition to your school, check out other programs such as Study Australia, which partners with over 21 Australian colleges, and Austra Learn, which offers semesters or years abroad at 30 universities. Finally, websites like Study Abroad list numerous resources and programs--231 academic programs in Australia and 66 in New Zealand. When examining these programs, consider when you want to go, how long you want to stay, and in which cities you would consider living.
Apply to the schools of your choice. Most of your applications will be filling in information and may not take long. However, you will have to request certain elements (like transcripts and letters of recommendation) well in advance--one month is best, though six weeks is better when requesting these items. In addition, most programs will require some form of application essay, highlighting your reasons for wanting to pursue the program of study. Focus on situating the study abroad experience within your own education. Discuss how the experience will provide you with new opportunities to learn and explore topics within your academic and personal areas of interest.
Renew your passport. This process can take a while (up to several weeks), so make sure that you take care of it well in advance. You can now do this by mail; the U.S. State department website has a list of requirements for submitting. If you have never had a U.S. passport before, visit this website to find an appropriate agency where you can apply for one (procedures will vary depending on when you are traveling and your history).
Apply for a visa. Check with your program's administrators for more information on what type of visa you will need, and how long you will need it for (this will vary by program). In general, Australian visas can be received electronically, online; New Zealand visas are available by mail. Verify your program procedures and requirements before applying for a visa.