Thanks again for all the suggestions - we checked out many of these options, including Turner's, TradeMe (which included lots of used car dealers advertising), and the Sunday car fair in Ellerslie. The latter seemed like a great source of cheap cars (under $5000) and a few in the low-to-mid range ($10k -ish). There was an onsite inspector you could pay for, as well as change of ownership services and such, even an espresso cart and hot dogs.
We ended up buying a 2010 Subaru Outback from NZC (which simply stands for New Zealand Cars...lol) in the North Shore for about $16k. We negotiated for them to throw in a new stereo unit with bluetooth and a backup camera, which were must-have features for us. We had a little leverage because we were interested in a somewhat more loaded car elsewhere, but it didn't seem like there was much of a wheelin' and dealin' dynamic at dealerships. Which was nice, honestly.
Source of payment was a bit tricky since we can't get a New Zealand bank account without a permanent address (seems that has tightened up a bit lately), and wiring money from home would have taken days because weekend/time difference/processing time. Someone on this forum said they used credit cards, but our dealer wasn't too keen. We ended up charging it all to our debit card, because apparently we have up to a $15k USD daily debit limit?? Who knew.