10 days would be just enough for one or other island but no way both.
Don't go north from Auckland, it's nice but a fair drive up and back and your time is limited.
From Auckland you should tour the Coromandel penninsula for 2 nights then around East Cape to Napier/Hastings for another 2 nights. From there come back west to National Park Village via Taupo and stay here www.howardslodge.co.nz
for 2 nights and walk the Tongariro Alpine Crossing (NZs best one day walk).
2 hours north from here is the Waitomo caves and 1 hour south the National Army Museum at Waiouru.
From Waiouru south to Wellington there is nothing to see but Wellington is worth 2 or even 3 nights but best make it mid-week when you are there.
A nice finish would be a ferry trip across to Picton on the other island for the day - return the same day or even stay the night. Fly back to Wellington from Picton 'airport' on sounds air shouild interest you.
Fly back to Auckalnd from Wellington.
I am from Picton and now live in National Park owning Howards Lodge but I'm not biased.
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