I've travelled most of both Islands, sometimes by campervan and others by hiring a car and staying in hostels/hotels. I'll start with the North and work my way down. In three weeks, you can do a fair amount, although if you're driving it will mean a lot of time spent on the road. There are areas, especially in the South Island, where you will drive for hours and not see another human being (plenty of sheep though!).
Another thing to bear in mind is that unlike the turn of the century, where it was $4NZ to the pound, NZ has become quite expensive to travel and reside in. I believe it is around $2 to the pound now. Things like groceries, drinking out etc can be more expensive than London, especially in tourist areas and larger cities. However, there is a lot that can be done for free in NZ, nature-wise, and if you are driving you will find that you'll stop every half-hour to forty minutes because a point of interest will be signposted.
Flights - the time you are going is the beginning of summer and heading into Christmas, so can be quite pricey. We ended up going through STA travel. For a flight to Dubai, Thailand, Sydney and then Auckland return we ended up paying around 770 quid if memory serves? They can get you bespoke deals and although we weren't students, they gave us student discounts. Might have just been a very nice rep we spoke to though. We flew out at the beginning of October.
Flight Centre tends to do good deals as well. We flew with Emirates the whole way apart from our flights within SE Asia, which we had to book seperately. Air New Zealand is a fantastic airline as well. If you end up saving money due to a long stop-over somewhere, I would suggest making it Singapore, most ridiculously brilliant airport I've ever been to and you won't get bored! They also have a fairly inexpensive transit hotel. I paid $60US for a 12-hour stay.
Weather-wise, the time you are going is when the sun starts shining. If all goes to plan, you'll hit temperatures of the late 20's-30's during your time in most regions. Always make sure you have warm clothing available though, the song Four Seasons In One Day is very factual! The South Island has areas that can get bitterly cold, even in Summer.
Transport - If you decide to campervan it, there's plenty of fairly cheap holiday parks around both islands to stay in, normally around $18 per night for a non-electric plot or just under double that for electric plots. Unfortunately, you can't just stop anywhere now, as you used to be able to do. Mainly due to pollution and litter problems. NZ is a very green nation and they don't take kindly to people abusing this. Campervan deals are not too shabby, depending on what type you want to get. The main rental places are Backpacker, Jucy (who also do budget accomodation), Spaceship, Maui, Britz and my personal favourite, Wicked, because of the graffiti inspired vans! Always try to ensure you don't go under a quarter-tank. In some areas of the South Island, fuel stops are few and far-between. Fuel-wise, your budget can take a hefty hit compared to getting a car and finding cheap acommodation.
Cars are a good option if you aren't fussed about the type of hostel/hotel you lay your head at the end of a long drive. They'll get you from A to B quicker although some of the terrain can be pretty rough.
Coaches are another great option but you just lose a bit of freedom to stop off at places of interest. The cheapest ones I have found in my time there are
http://nakedbus.com/nz/bus/, and then you can find local deals wherever you are for sight-seeing.
I never got round to doing this, but there is a train that goes straight from Auckland to Wellington. It's meant to be breathtaking
http://www.kiwirailscenic.co.nz/northern-explorer/. There are also scenic trains running across the South Island for various tours. The most famous one is the Tranz-Alpine
http://www.kiwirailscenic.co.nz/tranzalpine/?gclid=CJapoYWo4rYCFUlapQodlQkAJg.
People - Kiwis are some of the most amiable and helpful people you are likely to meet. Nothing is too much trouble and don't be surprised if you are invited round for dinner by the cabbie who dropped you off at your hotel, who just happens to live nearby! Don't be fooled by the size of the Maori's and Pacific Islanders, they are some of the most placid and chilled-out people you will meet. Although with that does mean you will experience 'Kiwi-time', where that thing you needed urgently will arrive at some point or other as and when they get round to it :lol:
Deal sites are your friend! We saved a lot of money booking things up on deal sites. You can book acitivies, short breaks, accomodation and transport as and when they come up. This is a good site to check out as it encapsulates all the major deal websites
http://www.dailydealsites.co.nz/
North Island
Cape Reinga - The tip of the north Island. You can walk right up to the tip and watch the Tasman Sea crashing into the Pacific. Beautiful scenery and very spiritual area.
http://en.wikipedia.org/wiki/Cape_Reinga
90-mile beach - depending on your transport plans, you can go on a tour of both this beach, some of the forestland around it and Cape Reinga in one day. This is just a pretty vast, straight beach up the west coast. Pretty awesome waves and scenery.
http://www.newzealand.com/int/feature/ninety-mile-beach/
Bay of Islands - Stunning area. The main town is Paihia and from there you can book in tours of the surrounding islands. There is also the opportunity to swim with dolphins in their natural habitat, skydiving and other such activities. Probably best to spend at least a couple of days here if you are interested.
http://www.bay-of-islands.co.nz/
Poor Knights Island - If you've got your PADI, great dive spot. I got mine from Global Dive and went on a few dive trips with them. Generally good value and really experienced dive masters.
http://www.globaldive.net/global-dives-poor-knights-liveaboard-weekend/
Waitakere Ranges/Piha Beach - not far from Auckland, a vast rainforest reserve and excellent and intimidating beach. Lot's of tracks and waterfalls. Can be done as a day trip out of Auckland.
http://www.aucklandcouncil.govt.nz/EN/parksfacilities/premierregionalparks/Pages/waitakereranges.aspx
Auckland - I lived in the city for two years, although it is the most populated area it is still tiny compared to cities in England. Fantastic little areas to chill out and for entertainment. Look up places like Ponsonby, Parnell, Devonport and the main city itself. Then you have the various domains, which are basically volcanoes within and surrounding the cities. Great views from One Tree Hill and Mount Eden. Very, very hilly, as it should be considering it is built on volcanoes! The main Auckland Domain houses the museum.
Would be worth checking out Waiheke Island for its lovely views and world-class wineries. I proposed to the missus here as well, very romantic! Then there is the the Viaduct and Wynward Quarter area, which is the main hub for going out right by the docks. Very expensive though. Karangahape Road is the sleazy, back-street Soho type place on one long-ish stretch of road in the city.
Rangitoto Island is a dormant volcano out in the gulf, you can get there by ferry, doesn't take too long at all.
http://www.aucklandnz.com/destinations
Coromandel Peninsula - Another absolutely stunning region (get used to this, and where not even on to the South Island yet!). Plenty of activities and places of interest. I spent New Years Day at Cathedral Cove one year, a small beach tucked away within the forests. It is a good 30-minute trek each way but well worth it.
http://www.thecoromandel.com/
Tauranga - Great beach city, general lovely sights etc and a dormant volcano to climb.
http://www.tauranga.co.nz/
Rotorua - Smells of rotten eggs (for reals) due to the thermal activity around the area. Lot's of traditional Maori events and areas to check out, bubbling thermal pools and geysers etc. Don't wear jewellery if you go into any of the pools (not the bubbling ones, you'll be burned alive!) and you'll need to give your clothes a good couple of washes after visiting there.
http://www.rotoruanz.com/
Lake Taupo - Stunning, huge lake with lots of lovely areas surrounding it to visit. Apart from Queenstown, the best place for your adventure activity needs, such as sky-diving, bungee, white-water rafting and so on. You can get deals on doing activities in bulk if you wish.
http://www.newzealand.com/int/lake-taupo/
Waitomo region - A must for the adventure seeker! Glow worm caves, blackwater underground rafting and we abseiled 100 meters down into a cave system. Fantastic time there! Great little hotels and holiday parks to stay in.
http://www.waitomo.com/
We booked most of our activities through these guys
http://www.waitomo.com/black-water-rafting.aspx
Tongariro - An absolute must. If you're a fan of LOTR you'll recognise the mountains. When we went we unfortunately couldn't do the crossing due to extreme weather and the fact the volcanoes had erupted fairly recently!! However, I believe they've opened it up now. Best to double check before you go whether you can get the most out of it. Ridiculous terrain, views and settings. You'll need to be pretty physically fit to attempt the harder crossings.
http://www.tongarirocrossing.org.nz/
Taranaki - beautiful drive around the base of the volcano and in good weather you can tramp up it. You can see the Tongariro summits from here and vice-versa.
http://www.taranaki.co.nz/
East Coast region - Areas like Napier and Gisbourne are world-renowned for their wine. Napier has various old Art-Deco buildings and Gisbourne is another very pleasant beach-side region.
http://gisbornenz.com/
http://www.hawkesbaynz.com/?gclid=CLftscis4rYCFYUhpQodzm0AAg
Wellington - the Capital city, about half the population of Auckland. Fantastic little place. Botanical gardens are great, there's a cross-section of little streets going off Cuba Street with awesome little bars, cafes and restaurants. Very vibrant and artsy. Then there is the Beehive, their Parliament HQ. Right by the water again, so lovely views. We flew into Wellington, which is an experience in itself. I am fine with flying, but there was a moment where I thought it was all over on the landing! When the weather is good, it's sublime but be aware that it is incredibly windy a lot of the time so best to have warm clothing with you.
http://www.wellingtonnz.com/australia
So that's the North Island in a nutshell. I've honestly just scratched the surface though and you will find plenty for yourself. I'll start compiling some info on the South Island as well, and that is where brick gets real! Plus some stuff on Sydney as well. Be happy to meet up when you're here if we're still around as well.
Hope this is useful mate, will send some more info soon.