There is a lot of information available online for cycling in New Zealand like the official site New Zealand Cycle Trail.
I only list the routes that I personally liked most.
There are a some Bike Trails in New Zealand. Most of them are made for mountain bikers, meaning they are mostly unsealed roads. Sometimes purposely made for cycling, sometimes quiet gravel roads. It is not always easy to find out how difficult they are. Most can be done with a touring bike too, but the more difficult ones are then certainly a challenge.
Generally I found the South Island much more interesting and more diverse. On the other hand there are more Bike Trails on the North Island. My favorites in the south are:
Molesworth Station & Rainbow Valley
Across the biggest farm in New Zealand there is a great gravel road. Scenically totally different then elsewhere with wide grassland without trees or fences. The ride through the Rainbow Valley is a perfect combination to get to the start of the trail. Check with DOC if the road is open.
The ride to the northernmost point of the south island leads over the Takaka Hill, one of the nicest passes for cycling in New Zealand. But the highlight is the wild Wharariki Beach with a rock arch and a seal colony.
A rather easy trail mostly through deep rain forest.
Queenstown (Walter Peak Station) - Te Anau (Around The Mountain Trail)
To the start of the road you only get with the steamboat from Queenstown. Because of that you have the beautiful and lonely gravel road almost for yourself.
There are quite some reasons for not taking this road: a narrow road with lots of tour busses and a 10% steep, dark tunnel. But the scenery along the way is some of the best, New Zealand has to offer.
Nice gravel road over a mountain pass in Otago.
Mt. Cook Village
The views along Lake Pukaki towards Mt Cook are just fabulous. Additionally there are great trekking options all around Mt. Cook Village.
1st part Alpes2Ocean Trail
To the start of the trail you only get with a helicopter flight. But the section from Lake Pukaki to Omarama is fantastic riding. The second part is unfortunately mainly a signposted main road.
Queen Charlotte Trail
Most definitely a mountain bike route, but it is possible with a touring bike, if challenging. Not ridable are the first 3km and the section between Torea Bay and Onahau Bay (here there is a road as an alternative). Often great views of the sounds.
Ninety Mile Beach
On the west side of the northern tip of New Zealand is this 90km long beach. Great, lonely ride along the beach. Take a break during high tides as the water will push you to the unridable soft sand during that time.
Great ride through the deep woods. Partly following an old railway line, partly purposely built for bikers. Lot’s of suspension bridges.
Te Urewera Crossing
Public gravel road between Rotorua and Wairoa. Great scenery and beautiful lakes.
Consists of different trails. The Fisher Trail is a great trail, mostly downhill (can be done uphill too). The Kaiwehakauka Trail leads through a wild and very remote valley. Done with a touring bike, certainly a challenge.
Mountain bike classic following an old logging road. Challenging on a touring bike.