Sending and receiving physical stuff Edit
The New Zealand post office is reasonably reliable and efficient. If you're sending stuff domestically, you're in pretty good shape.
International mail can be more of an adventure.
Receiving mail and parcels from international senders Edit
Tell the sender to get a tracking number. This may make it twice as expensive or more. That's annoying, but I have had stuff straight up lost shipping from the US to NZ, and on both ends they will tell you there's nothing they can do if you didn't get a tracking number.
Receiving stuff by courier Edit
The popular US couriers (UPS and FedEx) are not big here, and are usually used by businesses rather than individuals. You can use them, though it may be expensive. If the place you're living has security (such as a gate or a keycode-access door), do not expect that the courier will have access. I live in an apartment building in the CBD with a keyfob-access door. I saw on the UPS tracking page that delivery was attempted, but they left no note and did not try to contact me. If you call them and give them your number, in my experience they will call you when they attempt delivery though.