Probably the most straightforward way to begin is with INZ website factsheets about the various visas. For instance, here
https://www.immigration.govt.nz/new-...-resident-visa is the one about Residence through the Skilled Migrant Category - this is possibly the route that your wife will have to take eventually, by getting a qualifying job offer when she has finished her studies. You will see that the page has tabs for different aspects of what your wife needs to know, e.g. the Criteria tab lists the requirements, and has links to the Evidence needed in each case. Because of the long processing time for Residence, most people first apply for some kind of Work Visa to enable them to start working legally, while their Residence application is in the system. One possible is this.
https://www.immigration.govt.nz/new-...tudy-work-visa
Or this, which leads to a different route to Residence if it suits to be tied to the one employer.
https://www.immigration.govt.nz/new-...oyer-work-visa
Also, this
https://www.immigration.govt.nz/new-...nce-categories, involving the Green List (links on that page) is a new set of visas now coming into use - we haven't had anyone on the forum go this route yet, as it's so new.
https://www.immigration.govt.nz/new-...nce-categories and there is also a Work To Residence possibility.
Similarly, here
https://www.immigration.govt.nz/new-...lders#overview is the factsheet about the Partner of a Student visa. However, notice carefully the requirements for proof of a live-in relationship needed to qualify as a partner - it's not sufficient to be married. Here
https://www.google.com/search?client...of+partnership are old threads about proof of partnership, to show you the kind of thing you have to supply.
Note: there is no quick and easy way to understand all your options. Just at the moment, the NZ government seems to be revising requirements as they try to sort out the mess that happened because of the pandemic, so some visas which used to be available and that we were familiar with aren't there any more, and everyone has to readjust to the new possibilities. The INZ News and Notifications page
https://www.immigration.govt.nz/abou...-notifications is the place to watch to know about the latest tweaks.
Good luck.