What is the best way to travel in and around New Zealand?

In the main cities and towns there are good transport options - at the very least Taxis. 
Main towns/ cities also have buses. 
You can get Bus passes to travel between towns as well - NakedBus and Intercity are the two main ones. (They both have $1+ (plus booking fee) fares, so they're very cheap). 

You're likely to arrive to New Zealand in Auckland.  Auckland is a great city, you can spend time there. I prefer  Wellington which is where I now live. 

Auckland has a pretty harbour area in the city. 
Wellington has plenty of cultural, historical and interesting places to visit. If you like the Lord of the Rings you can also visit the Film set locations through tours that leave from the  Wellington Visitor Information Centre, these vary in price depending on where you go. 
If you climb  Mount Victoria you will also see some of the Lord of the Rings locations. Plus it has great views over the city. 

Cuba Street in Wellington is a wonderful (and awesome) street to people watch, drink luscious coffees, try delicious foods from cafes, restaurants and bars along the street. That is also where   The Bucket Fountain is. A great spot to spend some time. 

If you're after some history, culture and NZ life information head to  Te Papa Tongarewa - Museum Of New Zealand it's awesome. There are free things to see - and one "pay per view" area where the more fancy gallery shows are. (You can see the rest for free). Allow yourself at least 1/2 a day to see it, there's a lot to take in. Luckily there is a wonderful cafe onsite too. And check out the Art Gallery in the top levels - amazing art works! 

You can take the Ferry across to the south island from Wellington too - the Interislander is popular. https://www.interislander.co.nz/ 

It is cheaper to take the ferry without a vehicle. So you may want to leave the car hire at the  Wellington International Airport or other location for drop off.

I haven't spent enough time in the South Island to be able to tell you where to go. However...

Queenstown is popular for winter sports - skiing, snowboarding etc. However in January it's summertime for us, so you're more likely to be hiking and other walking type things. 

Kaikoura http://www.kaikoura.co.nz/ is quite a popular place to do Whale Watching. It's also not very far from where the Ferry enters the South Island - although it is still a drive. (Maybe a few hours, sorry, not sure, haven't done it. Yet!). 

Unless you like very slow travel, avoid trains (apart from the TranzAlpine which is one of the world's greatest train journeys). Car or campervan definitely the way to go. Big country and few people mean mostly empty roads and some great driving. Transit within the cities isn't great either. Auckland not so bad, but actually Wellington and Christchurch are samll enough to get around on foot or by cab if you'd rather not drive around.

Best way to get around New Zealand IMHO is 

A)helicopter (yeah, yeah I know... a little exotic), but its insanely beautiful. See below for proof.

B)rental camper van.  there are tons of these rental outfits around where you can cheaply rent an RV and see everything up close and personal.

I definitely recommend renting a car, or even a camper van, to travel around. Public transport is okay in the major cities, but options are very limited outside the centres.

The New Zealand country side is amazing (especially in the South Island) and there are plenty of little spots to stop with a car/van and appreciate the views, something you can't normally do when taking a bus or train.

I recommend driving from Christchurch to Queenstown. It was by far my favourite trip (although the first part until Timaru is a bit boring).

In saying that, there are a couple of nice train journeys worth checking out: http://www.kiwirailscenic.co.nz 

I second the renting a car option. I did the same trip as you, slightly longer, and though there were excellent taxi options in all the cities we visited, there always seemed to be a day trip just out of reach. Sure, for our trip to hike Rangitoto Island off Auckland there was ferry service, and from Queenstown to Milford Track there were tour buses, but we missed out on a lot of countryside, small towns and scenery by taking taxis and planes everywhere.

Hey! My recommendation is to rent a car, or even to buy a second hand one (you can find some really cheap ones) and then re-sell it when you leave the country.

Agree with most of the answers already but just want to suggest you to check out Transfer Cars for free one-way car rentals.

I would suggest the train.  Their rail system is amazing and you will save time - and money.

You can take the name bus or intercity which are great but it all depends what you want to see. Nz is a pan awesome country but sometimes, when you get somewhere the attraction can be a bit off the beaten track so requires local transport. I would recommend either renting a car or camper or traveling by bus to specific areas then renting a car locally. Trains aren't so well developed so I wouldn't count in trains to get around. If you do decide to drive, remember that you cover much less ground in nz than Canada because of windy roads  and overall slower pace of traffic. 300km willtake you bêtes 4 to 5 houris here.

The best way to see and experience New Zealand is by doing a road trip. It's a Kiwi right of passage. Whether you hire a car and stay in motels / hotels, or hire a campervan (like lots of tourists do) is up to you and your budget. Just be aware that a few of the smaller roads, especially in the South Island, aren't good for campervans (your campervan company will tell you which ones). 

Don't bother with trains, as they're slow and don't go to many of the places you need to visit. To get from Wellington (North Island) to Picton (South Island), take the Interislander car ferry. It's one of the most spectacular arrivals you'll experience (the Marlborough Sounds are amazing). Think islands and dolphins jumping off the bow.

2 weeks each island is a good amount of time. North Island must-sees include the Coromandel Peninsula, Hawke's Bay wineries, Tongariro National Park, plus Wellington's cafe and bar scene.

South Island must sees include Marlborough wineries, West Coast limestone rainforest,  Queenstown for extreme activities, plus Milford Track.

Public transport in Auckland is mostly bus-oriented, with a few unreliable trains. Wellington is generally better, but stay centrally and you'll be able to walk. Not sure about Christchurch, things have changed about a bit since the earthquake.

Hire a car and drive around. The scenery is breathtaking

I would suggest either hiring a car , or campervan. I think hiring a campervan is pretty popular, and allows you to see the outdoors a little more closely. You can stay near the national parks and by the beach. Definitely the way to go. 

My video should give you some great visual ideas! :) https://www.youtube.com/watch?v=qPsDfGyZ3zA 

We just got back about a month ago!