The process of getting a New Zealand visa is pretty straight forward in most cases
New Zealand has a very easy to navigate website that tells you what your options are for visas. Some visas require an employer to sponsor you, other visas like working holiday visas do not require employment.
When you decided which visa you are applying for it is time to start the application. If you are applying for a visa that allows you to be in New Zealand longer than a year you are required to have a health check with an approved physician. If your visa is for less than a year, this is not required. Your next step is to upload the supporting documents for your visa type and hit the intimidating apply button.
Typical processing time for most visas is about two months and immigration New Zealand lists the current processing time. When you submit your visa application it is first assigned a location that will process it. Many visa applications are assigned consulates in other countries (mine was assigned to the London consulate). Each case officer is assigned a certain number of applications at one time and when they complete one application, they are assigned another application. Your application will sit in the holding queue until all other applications ahead of it are assigned to case officers. Once your visa application is assigned an officer it is the first time that a human will have looked at it. It can take between a few hours and several weeks to get your visa application process completed once it is assigned an officer. If any additional supporting documentation is needed you will receive an email.
If you have questions about your visa application, I strongly recommend calling New Zealand Immigration. The call center is available 24/7 except from 12am Sunday to 6am Monday NZT and wait times are typically 60-100 minutes regardless of when you call. Unlike trying to call US immigration you speak to a real live person who actually has the ability to help you with a problem and is not limited to reading a script. The phone support has a liaison officer at each location visas are processed which they can communicate with about your application.
Most of the visas that New Zealand issues are eVisas. This is great because it does not take up a page on your passport and if you get a new passport you can notify New Zealand immigration and they can apply your visa to your new passport. Once you have your New Zealand passport you can start planning for your arrival in New Zealand.