Well, first of all you need to decide where you want to live (Australia? Or New Zealand? You might have trouble getting visas for both.)
Then you need to research the country you want to live in to confirm what massage therapist requirements are. If you're a LMT, that means you have a license in the US.... but not in any other country. So once you move there, you may need to get re-licensed in that country.
You'll need a work visa. Sometimes you need to have a "sponsor" in order to get that. If you can secure work prior to leaving the US, that company will probably sponsor you.
Finally, if you go without having a job lined up, you should go there with about $10,000 in your pocket. That will get you 3-5 months of getting situated just in case you have difficulty getting a job when you get there.