Things have changed quite a bit since COVID - both in terms of the demand for teachers and the immigration and policy requirements. You'll need to get registered as a teacher (after having your qualifications assessed, if necessary) and then start job hunting. There is only a limited opportunity to enter NZ at the moment for teachers, so you might not get a job offer for a while. It is (obviously) a very expensive process.
The big question will be whether (if you get a job offer and a temporary visa) you'll be able to get residence. The residence category is on hold right now, and it's likely there will be big changes to it when it re-opens. People who are here now (or who can get in soon on a Critical Purpose Visitor Visa) can get residence through a special one-off visa category. For people who arrive after the border re-opens, it is very uncertain how the pathway to residence will look. It is really important to bear this in mind if you are planning to sell up and bring your family over - it's awful to see families who have been on temporary visas for years but who just can't (for whatever reason) get residence.
Let us know how your consultation goes! Good luck.