I spent most of 2004 in New Zealand. I was lucky to have time to travel around and see this amazing country. I was recently asked by a friend to give my advice on what to see there, when you have about three weeks to travel around, flying into Christchurch and out of Auckland. So here it is:
Three weeks in NZ gives you a good deal of time for sightseeing. If you're hiring a car I'd hire it for the three weeks, get a company that is based in Christchurch and Auckland and you can leave it Auckland when you're leaving.
Christchurch is only worth a couple of days really. There is much more to see in the rest of the country and you need to save time for later. It is worth a wander around the town though. You can get a walking tour pamphlet at the tourist office in the main square which will take you around the main sites.
From Christchurch I would head south to Dunedin. The Moreaki boulders are on the way and are worth a look, to break up the trip. Dunedin is the main city at the bottom of the Island. Another walking tour is available here, from the town centre tourist office. This is another self guided tour which takes a few hours and will give you a good taste of the city. It is also worth getting out to the Otago peninsula, where you can see New Zealand's only castle and go penguin watching. Dunedin like Christchurch is probably only worth a couple of nights.
From Dunedin your next site to visit should be Milford Sound. I'd go here via Invercargill, which isn't very interesting, but the drive there along the Caitlin's is nice. The drive takes you near to Slope point, the Southern most point of the South Island and it also shows you what the unpaved roads are like in New Zealand and why the W.R.C. drivers rate NZ as one of the top spots for rallying. The road is rough but wide and very safe. You can stay in Te Anau and then head to Milford Sound the next day. (There are bus trips from here with a cruise included and this is probably the best way to see Milford Sound).
Next stop is Queenstown. Haven't been there during the summer, but in the Winter it is very busy, with some great skiing. It should be a bit quieter in the summer. If you're up for it you should do the "Thrillogy" bungee jumps (http://www.ajhackett.com) . They are fantastic. The 4x4 tours are interesting as well and you should also do the Shotover river jet boat (http://www.shotoverjet.co.nz) . One activity that wasn't really available in the winter there is white water rafting. Again worth it if you're up for it. Also definitely worth taking the gondola ride to the top of the hill and look at the lake and while your up there I would pay for five runs on the Luge.
You could spend a few days in Queenstown, allowing you to relax a bit after the driving so far. From Queenstown I would head for Franz-Josef Glacier. This is the better of the glaciers. When I went, we weren't able to get the heli-hike tour due to terrible weather, but it should be more reliable in the summer. This needs to be booked a few days in advance. If you can't get a helicopter to take you onto the glacier then a hike up the front of it is worth doing as well. On the way to the glacier you could stop in Wanaka, at The Puzzling World of Stuart Landsborough (http://www.puzzlingworld.co.nz) . This is an interesting stop with a museum style set-up dedicated to puzzles and optical illusions. If you have a bit of time it's also worth trying out the 3D maze there. I managed to do it in less than thirty minutes and then had to lead my parents to safety.
After hiking on the glacier I would start heading towards the North Island. The Tranz Alpine is a great train journey, between Greymouth and Christchurch, but it might be better to hold onto the car and cross the mountains that way. North of Greymouth is Abel Tasman Park, which is also supposed to be nice, but given the time constraints it will probably need to be missed.
When you get back on the east coast I'd start heading north again, maybe stopping on the way in Kaikoura to do some whale watching (http://www.whalewatch.co.nz/) , before getting on the ferry for Wellington. You can take your car if you want (http://www.interislander.co.nz) .
You can also do the first part of this trip in the opposite direction from Christchurch if you want. Start with the glaciers and do Dunedin last.
Wellington, as the capital of New Zealand, is worth a few days. The Botanic Gardens is nice and you can take a tour of the Parliament building, if you are there early. You can also sit in on a session. The National museum, Te Papa, is also definitely worth a visit. It's massive and worth getting a guide book which will give you a route to the highlights (http://www.tepapa.govt.nz).
Now it's time to see the North Island. From Wellington I would head for Napier, which is famous for it's Art Deco architecture. Another self-guided walking tour is available here and will give you insight into the rebuild of the city post a large earthquake. From Napier you can then head to Rotorua, via Lake Taupo. Skydiving is available at Lake Taupo and on a clear day it is a great view. The diving is done in tandem and is great fun (http://www.skydivetaupo.co.nz) .
Rotorua is based in a region of volcanic activity and here you'll see bubbling mud pools and geysers. There is also a pervasive smell of sulphur which takes a bit of getting used to. Other things to do here are the Farm Show and Rainbow Springs trout farm, which are worth a visit (http://www.rainbowsprings.co.nz/home.asp) . You can see a kiwi here, a bit artificial as it's in captivity, but they're very hard to see in the wild. If you didn't do the luge in Queenstown then there is another opportunity to go up the gondola to look at another lake and do some lugeing.
From Rotorua I would start heading towards Auckland, taking in the Waitomo Glow Worm caves on the way (http://www.waitomocaves.co.nz/home) . Plenty to do in Auckland, but if you have time left I would head North first towards the Bay of Islands. I would stay in Paihia. Keri Keri is a small town nearby accessible by ferry and it is one of the oldest towns in New Zealand. It's worth spending a few nights here to do the Cream Trip cruise around the islands (http://www.fullers-bay-of-islands.co.nz) , as well as a day trip to the North Cape along Ninety Mile beach (By bus, or if you're still flush with cash by plane, or if you have a couple of thousand burning a hole in your pocket by helicopter). You also need to visit Waitangi Treaty Grounds, where the treaty was signed and New Zealand was born. You'll be able to learn loads about the treaty and the difficulties surrounding it in Te Papa, Wellington.
Then it's back to Auckland. There's loads to do in Auckland. Kelly Tarlton's Antarctic Experience and Underwater World is a top attraction (http://www.kellytarltons.co.nz/home) . Also recommended are the ferry trip to Devonport and the Wild West Coast beaches, especially Piha. Then time to dump the car and leave the country.