You should first start by reading through the various visas available and other info, on the official Dept of Immigration & Citizenship site (see under Visas/medicals).
There's a severe shortage of all qualified nursing staff, so this diploma would be to your advantage.
The easiest visa to get is the regional one, which means you would be willing to go to a small town, outside of the capital cities, where the worst shortages of all qualified staff are.
You should be aware that it's a costly process, with no guarantees, or refunds, should you fail to be granted a visa, or change your minds. Your dogs will each cost the equivalent of a full airfare, including the transport and quarantine etc. So you'll be looking at approx eight airfares plus medicals etc.
Because of this expense, you need to be absolutely sure before starting the ball rolling.