OK, now you all are making things hard to decide. I will lobby hard for a South Island emphasis but we do have friends in Wellington that we definitely want to visit. Rotorua sounds like it has lots to offer, both good and bad. The trek through the Waimangu valley sounds intriguing too. Maybe we could do a sort of touch-and-go on the North and then concentrate on the South. Do any international flights (from the US) go directly into the South Island or is Auckland the only entry point by air?
From memory the only international flights that come through Christchurch are from Australia. But you could easily change to a domestic flight in either Auckland or Wellington.
Actually, we’re both Southern Men[sup]TM[/sup] so it’s all good. 
There are (or used to be) international flights from places like Japan, but yeah, generally to fly into Christchurch from overseas you’re going to have to change planes in Sydney, Brisbane, Melbourne, Osaka, or Auckland. 
Nothing interesting? How about Hanmer Springs, Arthurs Pass as a drive is amazing the Keas are incredible (if you get to see them)
For the north Island, I don’t think anyones mentioned Waitomo yet right? The glow worm caves are supposed to be spectacular (I haven’t been)
While I’ve never been there, Omarama in the South would be on my list. There you can get a flight in a sailplane (in and of itself a great experience) over some of the world’s most spectacular scenery. Its one of the “Meccas” for glider pilots like myself and I hope to go there some day.
Glow Worms? Oh that’s a must see. 
Then don’t bother. They’re vastly overhyped.
It’s great if you want to glide. Otherwise it’s a very small town in the middle of nowhere.
One of the most enjoyable experiences I had while living in New Zealand was the Taieri Gorge train trip from Dunedin.
Otherwise I’d suggest spending quit a bit of time in Wellington, which has a lot of great things to see and do in a very small area. And don’t bother with Auckland.
Yeah that whole area is nice. Tekapo, Pukaki, Mt Cook, and Omarama. The lakes around there are fed by glaciers and have a beautiful turquoise colour. The westerly flow over the alps creates good mountain waves which are great for gliding.
I lived in the Dunedin area most of my life, and have traveled through Central Otago and Canterbury a lot.
For some reason I have a lot of memories of Twizel. I’ve only been there twice, but it has stuck with me. So there’s a nice spot to drop by.
Ouch. That’s perhaps a bit harsh. 
I mean, I wouldn’t make them the focus of an NZ holiday by any means, but if you’re heading from Auckland to Rotorua then the Waitomo caves and glowworm grotto are probably worth the side trip.
I was there in 2008. There was a fantastic restaurant in the town square. By far the best of any we went to in New Zealand.
My opinion is that 2-3 weeks isn’t enough time to do justice to both islands. Choose one & enjoy! & even though I’m from the North Island I would still choose the South Island of our two. It is just so stunningly beautiful 7 underpopulated.
If You do decide to visit the North Island there is still plenty to see & admire. I will join the chorus who have already said don’t spend too much time in Auckland (even though I grew up there.) The War Memorial Museum is worth a look though if for some reason you do get stuck there. & don’t visit Hamilton for any reason! Deadly dull.
Guanolad has given you a very good route.
Just to amplify slightly - Waitangi is extremely important historically & in a very beautiful setting. The Coromandel has some of NZ’s (& therefore the world’s ) most beautiful beaches. I’m especially fond of Hot Water Beach & Hahei. Napier in the Hawkes Bay is one of the great art deco cities of the world. Rotorua is world famous for its geysers & hot water pools- however, the smell can take a bit of getting used to. I don’t know the Taranaki well - but that is the part I would rush through if running short of time. I love Wellington - some good museums although I find Te Papa a bit theme showish.
Pak n Save is the cheapest supermarket chain overall. & the North Island of NZ features some shocking drivers!
I’ve mentioned those things, and as you should know from living in NZ, they’re more than a bit out of the way of the main Picton-Christchurch road. 
So my suggestion would be to fly straight to Christchurch from the North Island, rent a car, drive to Hanmer Springs and/or Arthur’s Pass on day trips, and then drive down to Queenstown and enjoy the scenery.
I’ll just add another couple of things.
If stuck in Auckland & you want to see a black sand beach go to Piha. Very beautiful - but very dangerous. Don’t be tempted to swim there - especially if the surf lifesavers aren’t on duty (Hot Water Beach is also very dangerous) Raglan (near Hamilton) is billed as a black sand beach, but last time I went there the sand is now dark brown - no idea why. Lovely town though & one of the few coastal areas that you will find an affordable camping ground.
Another vote for Waitomo to be added to your route.
Taupo is the North Island home of adventure tourism if you are wanting to try bungee jumping.
Rotorua is probably the best place to see Maori cultural performances. Someone from the area can advise if they are any good. (I’ve heard not that wonderful & overpriced) Maori TV around 6pm often has performances from various competitions & that might be better & save you some money.
Cafes - my favourite in NZ is the Replete in Taupo. (disclaimer; I know the owners slightly) Good cafes in Thames are Brew (in a historic old hotel) & the vegetarian Sola. Chequers has a lovely outdoor area if you are there on a sunny day, but I have found the service & food a bit hit & miss.
Food chains - Hell Pizza is overhyped & priced, but still better than Pizza Hutt & Dominos. Burger Fuel though is just great! If cooking for yourself or in a restaurant, make sure you try kumara (theNZ sweet potato)
& just a word of caution. Most “Kiwis” are wonderful people - but still use the same caution you would anywhere. Car conversions & thefts from cars are a big problem here - especially in car parks of well known tourist spots.