Australia and New Zealand signed the Trans-Tasman Arrangement, and according to that accord, the nationals of both countries can visit and stay in the other country with nothing but a passport. There are very few limitations. You could say that the Trans-Tasman Arrangement is some kind of European Union. As an Australian, you are free to move to and within New Zealand as you like.
The same rules apply to Australian residents. It does not matter what nationality you are, but as long as you have a resident visa for Australia, you have the same freedom of movement as an Australian citizen. You just need to make sure that your resident visa is valid. Otherwise, you will be the subject of the rules meant for your country of origin.
As you can imagine, there are a few exceptions as well. One of the most commonly met situations is when people are in Australia on a temporary visa, and they wish to move to New Zealand. In that particular case, you need to pay a visit to the New Zealand visa application center in Sidney, provided you are located in Sydney, of course.
If you find yourself in the situation mentioned above, you need to take all measures necessary and apply for a New Zealand visa. You cannot do that online when you want a work or resident visa. The online application process is available only for the Visitor Visa. Well, at least for now.
The bottom line is that Australian citizens do not need to pay a visit to the New Zealand visa application center in Sidney if they wish to visit New Zealand, work, or live there. The Trans-Tasman Arrangement makes sure of that. However, there are a few cases in which you need the center’s services, which is why depending on your particular situation, you should give the center a call and ask all the right questions. The information you find online should be used as a reference only and not as a 100% reliable source.