One thing you must remember, as KEN pointed out in the sites, NOT ALL the states require the same documents, to purchase and transfer the vehicle to your ownership.. BUT, YOU MUST provide PROOF OF OWNERSHIP in order to ship the vehicle, and most of the time that requires you to spend at least half a day at the local DMV (Department or Division of Motor Vehicles), and often you must pay tax on the transfer, (which is non refundable), unless you sign an affidavit that you are not going to operate the vehicle. Some states you can get a new TITLE while you stand there, others will amil it to you, only after they have done an investigation to ascertain that the title is clear.
Since you are desiring to drive it a few weeks, prior to shipping it out, you must plan on a few hundred extra dollars to transfer the title, tax, and fees to license it. Insurance can quite often be arranged through your local agent in NZ before you depart, if they are a large company. (Similar to the "traveling card" you can get to indicate that you have insurance when you go to rent a vehicle. This indicates when it expires and might also indicate the limits of coverage. )
Also as previously discussed, most juristictions require you to have a 6 month policy, but you can get a refund on a policy once you can prove that it is no longer in their jurisdiction. You send/take a copy of the policy, along with shipping documents indicating that it has been shipped to New Zealand permanently, and you'll get a checque in the mail, not directly proportional to what you would expect, but a refund. (If you have a clean record, and with the powers of computers they find out quickly if yo don't, you might have to shell out anywhere from $250 to $1000 for a six month policy.
Back to shipping, lots of pictures when you turn it in, (include oddometer statements), so you can get good coverage from the carrier of the vehicle, in the event the container is lost overboard in a tsunami or some knucklehead smacks it up "hot footing it in the ship yard" etc.
Shipping a vehicle will be best expedited through a brokerage, as I believe that Southern California to New Zealand, there are a few lines which go that way, but not too many which routinely carry cars. So your vehicle would be "containerized" in Los Angeles, loaded on a ship heading to New Zealand. There is scheduled service that way. (Containerization has it's positive sides, but provide desicant sacks to absorb extra moisture in your interior.) Not all containers are tight nor is the trip guaranteed to be a smooth one for your new acquisition. Also putting stabilizer in the gastank to keep the gas from seperating. (Most lines require the vehicle have no gas in it, but you must have something to keep the moisture from rusting the interoir of the gas tank, and evaporating and all the "gunk" clogging up the filter on the fuel pump, and the regular fuel filter in the line.
I know, too much too soon, but having shipped half a dozen vehicles while I was in the military, I've learned a few things.
Lots of Luck!!!!