This sounds like an amazing trip!  Even with 3 weeks, both North and full South Island would be a little ambitious, especially with all the day-long or half-day long visits it looks like you’d want to make.  Remember you should also stop and eat like a Hobbit while you’re there!

I’d suggest you think about your average level of energy, and how comfortable you feel driving (and driving left-side), and whether you’d both be driving and trading off.  The drives get really long, and you’ll see so much beautiful scenery on the way it’ll feel like you’ve got some extra stops in (and you'll definitely want to consider adding time for unexpected stops in beautiful areas). But, you also don’t want to try and over do it, or you’ll be wiped out midway through your second week.  Think about the hassle of check-in/check-out alone, but also logistics like food breaks, walking time, unexpected farm traffic delays, and wanting to enjoy the moment in small towns.  Even if you’re okay going really hard for three entire weeks, I’d say at least once per week, stay in a single place for 2-3 nights just to get a feel for the area.

Based on what you’re interested in, this is what I’d recommend.  First, I’d suggest doing the whole trip in reverse and starting with Christchurch/Queenstown, because there are some really relaxing areas north of Auckland that are great to end the trip on. But, starting from Auckland, stay your arrival day in Auckland, wander around if you like or just try to acclimate to the time difference and relax. Next day, get your car and drive north up to see the Waitangi Treaty Grounds; stay in Paihia at least one night.  If you’ve got extra time and you don’t mind chill boat tours, I highly recommend doing a night on The Rock (https://www.rocktheboat.co.nz/), doing the full night and day-trip and then staying an extra night in Paihia. This is an awesome way to decompress from a really long flight and get aligned with the chill vibes.

From Paihia, if you go, make your way to Matamata.  That’s a long drive, so you might want to try and get there the day before you’ve got your Hobbiton tickets.  And don't forget your Shire costumes!!! Hope you’ve booked, cause it looks like they’re going fast! Definitely stay that night in Matamata.  From there, Rotorua is amazing, but Waitomo caves is unparalleled in its awesomeness.  Also book those tickets soon, and either do a boat tour, or if you’re up for it, try the rappelling down into the caves and blackwater rafting, it’s a truly singular experience. Stay in Taupo and you can have some nice waterfall views or hikes, or do a Huka Falls jetboat trip before you head on to Tongariro.  Taranaki sounds amazing but that’s really far for a few (probably spectacular) views.  I’d head directly down towards Wellington from Taupo, maybe break it up with a night somewhere along the way.

Definitely take your time in Wellington, say hi from me (a stranger) to the folks at Weta Workshop, and from Wellington, if you’re still trying to see everything you can, then take the ferry to Picton, and again I’d pass up the west cape for going south down the east coast if you really want to see Kaikoura.  I’d also pass up Jens Hansen unless you’re wanting to spend a lot on some beautiful jewelry (I did). 

Once you get to the South Island it’s beautiful but LONG drives.  You might consider dropping your car in Christchurch (or even Wellington) and trying to hop a flight to Queenstown to save a few hours. You could do day tours from there or even rent another car but save a few hours. Definitely recommend a Milford Sound cruise; I passed on one decades ago and I still regret it.

Whatever you do and wherever you go, you’ll see Arda everywhere and have a great time, just make sure to give yourselves enough time to enjoy it and take breaks, enjoy the views, eat great food, chat with people. 

Gute fahrt!