I'm fortunate that my work has meant that I've travelled to all sorts of out of the way places in NZ.
Must sees in the North Island...hmmm..
Most people will arrive into Auckland. There are a few fun day trips from Auckland - Waiheke Island for wineries, Rangitoto Island for walking and city views, previously mentioned Warkworth for small town life not far from the city. Devonport is short ferry ride from the downtown city. Lots of great places for eating out in the inner city (not necessarily downtown though).
From there...well most people tend to come to NZ for the natural beauty, so you could either head north for the Bay of Islands, or head south for Rotorua, Hobbiton, Waitomo Caves, Taupo. Less typical for a short stay tourist but where the locals go would be the Coromandel Peninsula.
If you have more time Napier is unique for it's post 1931 earthquake build - more wineries. If you like remote north of Napier towards Gisborne (and beyond if you have time) is empty of people.
The Central NI has vast areas (relatively speaking!) of volcanic plateau. If you're energetic the Tongariro Crossing is a popular day hike up a volcano - but be prepared!
Wellington is our hipster capital and where the government is based. It's has a fun, walkable city centre and often overlooked. Te Papa the museum of NZ has some interesting exhibitions always on the go.
Highlights of the South Island.
Ideally, you'd arrive by ferry from Wellington and travel through the flooded valleys of the Marlborough Sounds (lots of little bays accessible by boat only, native forest/bush, interesting history, dolphins).
Blenheim is a winery hot spot. Home of the Marlborough Sauvignon Blanc. Otherwise a bit of a sleepy town.
Nelson is the gateway to the beautiful Abel Tasman national park - to really appreciate this you need to take a boat up the coast to take a walk in the park - longer if you can. If you have time, heading further north into Takaka is less visited but has more amazing beaches and not a lot of people. Pupu Springs are amazing clear blue water.
Then the choice is either West or East... (or both). The west is green and bush covered and remote and rugged. The east is dry and farmed and more populated. There are lots of walks short and long on both coasts and through the passes that connect the two coasts.
Kaikoura on the way south is where you can do whale spotting tours. It's a sleepy town, but can be quite beautiful in winter when there's snow on the mountains that almost meet the sea. I've always wanted to stay at the treehouses at Hapuku lodge.
Christchurch is awesome (my home town), but is probably more of an interesting stop off point than a destination in it's own right. Unfortunately still famous for being a post devastating earthquake rebuild zone. Riverside market is a new popular hot spot in the town centre. Akaroa is my favourite day trip - the french history is interesting and the views on the drive over the hill are beautiful.
Heading further south on the way to Queenstown you'll drive through the Mackenzie basin - no people, lots of sheep and cows. Lake Tekapo and Pukaki are beautiful on a sunny day - reflecting the southern alps in the seriously blue water. If you're there on a grey day you'll wonder what the fuss is about. Fairlie is a small town with a world famous in NZ pie shop.
Queenstown and Wanaka are where all the best and worst things of tourism happen. Lots of adventure tourism at both places, plus wineries. Those that know best will tell you that Qtown isn't what it used to be...but it's still pretty damn beautiful any time of year. Wanaka is less populated, so considered a bit of secret by some people, but honestly it's not a secret.
I'm about to run out of text...but Te Anau, Milford and Doubtful sound are the next big tourist spots. The hype is deserved.
Good planning website: