Some flights to New Zealand include a stop off in Los Angeles, so as well as being prepared for entering New Zealand you need to ensure that you are following the required procedures to transit through LA as well.
Your passport must be valid for at least three months beyond the date you leave New Zealand. A visa is not required by holders of UK passports, as long as you are described as a “British Citizen” on the personal information page. Holders of any non-British passport should check with their relevant embassy to find out about requirements before departing.
Your passport must be machine readable for transit through Los Angeles. Passports issued on or after 26 October 2006 will have to include biometric data for transit through Los Angeles. British machine readable passport holders who are also British citizens should apply for an ESTA under the Visa Waiver Programme (please see below). Forms regarding the transit formalities will be handed to you on the plane, so make sure you have a pen with you. Anyone who does not hold a British machine readable passport, or who does but whose status is anything other than that of British citizen should be especially sure to check transit requirements before travelling. A machine readable passport can be identified by the presence of two lines of text as letters, numbers and “<”s, printed at the foot of the personal identification page at the back of the passport. Further information is available at www.usembassy.org.uk.
Travel to the US under the VWP will be subject to eligibility being granted by an application under the Electronic System for Travel Authorization (ESTA). An ESTA will also be required in order to transit the United States. If you are a citizen of a VWP country you will need an ESTA, regardless of the country of departure for your journey to the US. The ESTA application is a web-only system. A third party, such as a relative or travel agent, may submit an ESTA application on your behalf, but you are still legally responsible for the answers submitted. You should apply for an ESTA via the website. The response should be almost immediate. You are highly recommended to apply for your ESTA as far in advance as possible, because if you are for any reason denied an ESTA you will then need to apply for a visa. The ESTA currently costs $14. An ESTA is valid, unless revoked, for multiple travel to the US for two years or until your passport expires. It is particular to the passport mentioned in the application, so a change of passport means a new ESTA application. Amendments to destination addresses or itineraries, for your original trip or for future trips, can easily be made via the website after your ESTA has been approved. An ESTA does not guarantee your entry to the United States, which remains subject to the usual inspection by customs/immigration officials on your arrival. You can find out more information about USA entry requirements at the Government Services and Information website.  
Transit arrangements are very strict in the USA, so please be patient and expect to queue more than once during your transit through Los Angeles airport. Be sure to take the normal precautions with your valuables that you would in any large city.
You can find out more information about New Zealand entry requirements at the Government Services and Information website.
No sharp objects, including pen knives or nail clippers, scissors or files, must be taken into the plane cabin. All liquids may also be refused in cabin baggage, unless you can prove, with appropriate certification, that they are for essential use (please check your airline’s documentation for its specific policy on this). Such items will be confiscated unless they are in hold baggage. 
Also see Taxes & Customs.