Wow, huge question if you are just starting. Firstly as you will realise, jobs are key to successful migration. I'd start by looking at requirements for teacher registration and also as an Engineer. (see
www.teachnz.govt.nz and ipenz.org.nz)
very generally speaking, if you had job offers in Education, or Engineering, then you would most likely get visas to work in those areas. Such visa might take 4-6 weeks to be processed in London. You would need police clearances and medicals if these are permanent roles.
A residence application, which has a job offer included might take 3-4 months to be processed in London, but doing that is a long way down the track.
If your are job hunting see
www.seek.co.nz or edgazette.govt.nz
an sorry, don't understand what you mean by "express entry" it not a term we use in NZ