This is very risky. It is expensive. The best way to try and achieve this without the risk is to rather obtain a skill in South Africa that is needed here in NZ and not take this risk.
If you still are very determined farming is a possibility:
1. Search the farm jobs at trademe - here:
http://www.trademe.co.nz/Browse/Jobs/Listings.aspx?mcat=5000-5015-5016-
2. Save for a return air ticket - book it - come by yourself and not with all the family (they can join you later if successful) - book your bus fares - save enough money for 8 weeks - work out your accommodation expenses here - http://www.bbh.co.nz/
3. When you are booked for this holiday - then apply for farm labouring positions starting your email off like this:
"I will be in your town on the Tuesday the 17th of August while visiting NZ on a holiday. I would like to look at the possibilities of living and working in NZ and may, if successful look at this as a real possibility given the opportunity. You have advertised the position of a tractor driver (or whatever)........etc".
4. If you get an offer of a job and you decide to try and live and work in NZ you can then apply for a work permit with the assistance of the employer, if they are willing.