The Southward Car collection is stunning - they also have probably the best Ruston traction in the world.
https://www.southwardcarmuseum.co.nz/
Colin and Esme Stevenson's collection at Tokomau is a stunning collection of engines and stationary engines but i'm not sure if it is still open to the public following their passing.
MOTAT is a good collection of museums to visit
in Auckland.
And as noted the Fielding club is fantastic collection of engines run by commited bunch of volunteers.
The paddle Steamer Waimarie in Whanganui is also well worth visiting.
https://www.waimarie.co.nz/
Napier is an Art-Deco town on the East Coat of the north Island and quite often has classic-car meets if you can coincide with one.
There will be lots i've missed but they're the ones that spring to mind.
Thomas.