Hi - There are a lot of great guidebooks and websites on New Zealand. I got overwhelmed of all the things you can do and experience. I ended up taking a Mondumo Tour (https://mondumo.com/). If you look at their website you can get some ideas for your own tour, of places to see and do. -- You might want to think about spending time on one island vs both islands unless you don't have the time and want to see a little of everything.
The North Island -
Auckland was great. The Auckland Museum is wonderful starting place. An introduction on the evolution of the country (plants, birds, animals, and people). Taking a ferry to one of the nearby smaller islands from Auckland is also fun.
Hobbiton - is interesting. I have not seen any of the movies or read any of the books but it was an interesting tour.
Rotorua - Immerse yourself in the Maori culture. Interesting. FYI - Due to a lot of geo thermals in the area, the town smells like sulphur. There is also a park that you can walk in the Redwood Forest. That was cool.
On the South Island -
Oamuru - cool town with a chance to see the Little Blue Penquin at the end of the day come out of the ocean.
Queenstown - a lot of adventures to be had here
Milford Sound (on the Tasman Sea) - only way to get there is by bus.
TranzAlpine Train from Christchurch to Greymouth - That was scenic.
Christchurch - So much to see and do. The International Antarctica Center is within walking distance of the airport.
I would fly Air New Zealand if you had a chance. I flew Air New Zealand down and a US based carrier on the flight back up. A huge difference. Also, if you have the funds, I'd fly premium economy. A little more room than economy.
Kia Ora!!