The best museums are"
The RNZAF Museum at Wigram, Christchurch
http://www.airforcemuseum.co.nz/
Omaka's Aviation Heritage Centre as noted above
Museum of Transport and Technology
(they have not run the Lancaster engines in years, sadly, it's indoors and fully restored now though. Also worth seeing the Solent and Sunderland, RB-34 and much more)
http://www.motat.org.nz/
Don't forget the RNZAF Ohakea Museum if he's passing there, ut's worth a look (just above Palmerston North)
Also be aware that Pioneer and Avspecs mentioned above are businesses, not museums, so prior arrangement for a tour is probably the best option. They, like all the other people at Ardmore Aerodrome, are very nice and welcoming if you approach them beforehand. Don't forget the NZ Warbirds are based on the same airfield.
There is also Ferrymead Museum in Christchurch which has some aircraft on display too, among a lot of other things.
http://www.ferrymead.org.nz/societies/aeronautical.html
And the Classic Flyers Museum at Tauranga is also excellent.
http://www.classicflyersnz.com/