There movement of people between Australia and New Zealand has always been very easy, our governments work hard to keep it that way while ensuring the biosecurity of each country.
I was recently waiting in line at Christchurch airport in New Zealand and the tourist(Australian) in front of me was severely reprimanded because his golf clubs and shoes where covered in pretty fresh dirt (from Australia). He got a nice little fine for his lack of cleanliness. This points out that while you can get some pretty cheap flights and travel deals to go to NZ, we still need to be vigilant to ensure no foreign (Australian) pests make the journey with us or our pets.
For dogs and cats New Zealand has no quarantine1 period when heading there, and there is no quarantine (in most cases) period to return to Australia either, but there are a number of important preparations that need to be done to ensure that your pets travel experience is as easy as yours is.