Australia has a complex visa system based on points. Your application is judged on many factors including your skills, your age, your English proficiency and can take quite some time (many months to possibly years to process). There is also a sponsorship work visa, which is usually granted within a space of weeks, however I have not heard of one being granted in conjunction with Chiropractic. For short term visitors, there is a 12 month work/travel visa which alklows up to 6 months of work, but (I believe) no more than 3 months with one employer. On a 3 month tourist visa you are not allowed to work. If your partner / spouse is Australian, then your relationship may be recognised for you to gain a visa. Ultimately it is very complex and we recommend using a reputable immigration specialist, based in Australia to consider your needs.
Registration in Australia recognises qualifications from many other Chiro schools, and it is mostly a form filling formailty. You are required to have Professional indemnity insurance, which can be taken out independently, or through the CAA which may provide for a significant discount.