A lot of people who come to New Zealand do so on a long holiday or working holiday or some other option that enables a long stay. If that's the case, then there is a thriving second had car market in all the major cities throughout the country. In the South Island, Christchurch is the main one and at any given time of year you'll find a lot of adverts up in every hostel and in various shop windows and internet cafes offering second hand vehicles. The public transport system on the South Island is a little sketchy and becoming increasingly expensive, so if you want to have a degree of freedom in where you go and when you go there then picking up your own ride is a very good idea.
During our trip, we bought privately and it worked out pretty well. We got a good car pretty cheap that served us well through the nine months we were there. To be on the safe side, if you do decide to go down the private route it's a very good idea to get an AA checkup (they cost about $100) to ensure that you know exactly what you're buying. You don't want to get 20 yards down the road and have the axles drop off, so if you don't know how to check a car out yourself then it's advisable to take this option.
An alternative is to go to one of the second hand car markets that the city has to offer. There are two major options in the Garden City and I strongly advise that you go to this one. When we were selling our car we went to both. The Backpackers Car Market, which is a little further from the city centre on Battersea Street, were shockingly bad. I brought in a good car at the last minute for a service they expressly advertise and they basically tried to pull a fast one on me. I had called the previous day to sort this all out and they said it was no problem, just to bring it on in and all would be well. When I got there two of them came out with me and spent the next ten minutes trying to fast talk me around to letting them call around and try to find me a buyer. To cut a long story short, they were trying to scam me. I later discovered that the people who own The Backpacker Car Market also own Lichfield motors (ironically a company which had previously scammed me when I needed some work done on the car), and they mess sellers around for a week or so then come in with a stupidly low price, supposedly off the street, and then run it through this second company.
This may all sound confusing, but suffice to say, they're con artists. They charge a little less than their competition, but I strongly advise you to avoid them.
By contrast, Christchurch Car Market (on St Asaph Street) were outstanding. I arrived at the last minute, the morning of my flight, they handled my query within five minutes, took the car off my hands and sold it for $1100 two days later on my behalf. The money was transferred through to me immediately. Also, looking briefly around the lot, the vehicles they had on offer were far more reasonably priced than the ones over at the Backpacker place. They charge a little extra to sell your car for you ($350 commission rather than $250), but they're easy to deal with, professional and honest. I can't recommend them strongly enough.