Since other people might also be interested here are a few suggestions:
Most cities have a model railway club with a track.
Auckland has the Museum of Transport and Technology (MOTAT) which has a beam pumping engine and plenty of other stuff to see from trams to aircraft.
Glenbrook Vintage Railway is not far out of Auckland, they have their own dedicated branch line and run steam locos. Would pay to check when they run first.
Whanganui has a working river paddle steamer, the Waimarie. Again it would pay to check when they run.
There is a good steam museum at Tokomaru near Palmerston North, however it has only just been bought by new owners and I am not sure what the situation is with visiting yet.
Near Christchurch there are two places, Ferrymead heritage park and Mcleans island "Steam Scene." I haven't been to either for a long time but I believe they are both worth a look.
At Ashburton there is the Plains railway, they have a beautiful Rogers 2-4-2 of 1877 vintage among other things.
There is a gas works museum in Dunedin.
On Lake Wanaka is the twin screw steamer Earnslaw, well worth a trip both for the scenery and the ship. The Earnslaw runs daily.
I don't know how you plan to travel within NZ, but many tourist find that a self contained campervan is a good way to go. Being a relatively sparsely populated country, public transport is not all that prolific although there are plenty of flights between the main centres. I think sometimes people book two separate campervans, one in each island, since it saves the cost of crossing the strait with one. We drive on the left here, same as the UK. Apart from near the cities, there is very little motorway and most main roads are only one lane each way. You generally don't need to plan to drive very far in any one day.
regards
John