Auckland is your most likely port of arrival to New Zealand as it's the country's busiest international airport. You could also fly into Wellington, the capital city at the bottom tip of the North Island.
For a major developed country, New Zealand is one of the most remote destinations on earth & getting here by plane can involve a lengthy journey. If you are coming from Europe, you can choose to fly either way around the globe & that will depend on price, timing, stopovers & plane changes. Also whether you want to see other places on the journey. The general rule to try & minimise jet lag is to follow the sun, which travels East to West; so fly into New Zealand from the US & fly out across Asia. Of course this is not always practical, but there are many other tricks that are worth researching, like always adjusting your watch to the local time & behaving accordingly, keeping very hydrated & waking up to bright sun if possible.
Flights to New Zealand have dropped in price in recent years with several major new carriers helping to open up the market. Major routes in and out of New Zealand come from Los Angeles & San Francisco on North America's west coast, Singapore, Hong Kong & Bangkok in Asia, Dubai or Abu Dhabi in the Middle East, Moscow in Russia, London, Frankfurt & Amsterdam in Europe and, of course, Sydney, Melbourne & Brisbane in Australia.
When you arrive you pass through duty free stores, but it is probably best to shop on your way out of the country, unless you want to carry everything with you during your stay. You need a passport and a completed passenger arrival card (which is best to fill out while you are on the plane) to get through passport control. It is also best to have a hard copy of your visa with you, if you need one. It is unlikely you'll get too many questions here, but if you do they will be focused on your travel plans & funds; where are you staying, what are you planning to do, do you know anyone here, how much cash do you have… All designed to ensure you enjoy a legitimate holiday for the right period of time.
If you are lucky enough to have an e-passport from the UK, Australia or New Zealand, you can pass through the automated SmartGate which will scan your passport & then your face to ensure it matches the passport photo. Wherever you are from, you shouldn't be carrying with you more than 3 litres of spirits, 6 bottles of wine or 50 cigarettes (or 50g of tobacco products) or you'll be charged tax on excess amounts. The detailed official list can be found here. If you are bringing in more than NZ$700 worth of goods (for gifts, for example) then you will need to declare them. And although you can bring in as much cash as you want, if you carry in more than NZ$10,000 (or the equivalent in other currencies), you will need to fill out a declaration.
Customs and Biosecurity are the final hurdles before you can enter New Zealand. Customs is very specific on what you are not allowed to bring into the country, primarily to protect the environment, so Biosecurity is taken very seriously. On the Passenger Arrival Card you will have declared anything you may be carrying. If in doubt, declare it! If you have something that is not allowed, you will have to either surrender it or pay a tax. If you do not declare something you should have, a heavy fine is always applied. Used outdoor equipment and footwear are the most common red flag items as they can carry unwanted microbes into the country. Check here for the latest, full list.
During your stroll through the airport after collecting your luggage you will most likely encounter the sniffer dogs. These usually adorable creatures have been trained to sniff out a huge variety of items, so don't expect to be able to smuggle hidden items in. They can detect fruit residue in an empty bag that has not had fruit in it for months! After your luggage has been scanned, you have officially arrived in New Zealand. Welcome!