Good question – this may depend on the type of business and the various regulations that relate to your trading. For tax, you don’t have to register, albeit you will have to record your income and expenses and report that. If you are less than $60k per year, you don’t need to register for GST, but you will need to use your IRD number for reporting. If you have a premises even home and you have customers visit you, you will need to register for a NZ Business Number (google that) so that you can get one of those QR Code Posters for people to scan in on the Covid-19 tracer app. Otherwise, if it is food related you will have various regulations that you need to cover off.
Hope that helps.