New Zealand food
I was there 7 years ago so the names of places does not matter today. But in general I suggest finding some wine bars that serve food. Reason, the wine in New Zealand is overall much better than the food. Find some good sauvignon blancs to eat with some goat cheese and you will be in heaven. In general seven years ago any place that seemed to have a good wines by the glass menu tended to have some good food too. Avoid the traditional english style food place (roast beef, puddings and such...)
The previous poster is right about a) New Zealand sauvignon blancs being excellent (anything from the Te Mata vinyard is good!) and b) the "English food place" in Christchurch being bad. The place he/she is talking about is named the Oxford on Avon - it's a big Britophile tourist trap, and the food is quite mediocre (I ate there. Once.) Also, avoid a dessert place called Death by Chocolate - I wasn't impressed, very gimmicky and the desserts lacked that je ne sais quoi.
A restaurant called Dux Delux, near the Arts Centre, is reliably good. A friend who lives in Canterbury recommends a place called Hay's (see attached link) and I have to say, this looks like one of the most tasty NZ-ingredient based menus I've EVER seen. And I live here (going on 3 years).
The streets around the Arts Centre in Christchurch are clustered with excellent restaurants, and on Saturdays and Sundays at the Arts Centre, there is a convocation of extremely interesting international food trucks providing cheap, tasty street food - Indian, Thai, Eastern European, baked potatoes.
If you have lots of money and want to spend it, eat a crayfish. It's the local version of a lobster. Try a local meat pie (ask if it's homemade) for a budget snack. Any bakery or sandwich shop will have a selection of retro British-style baked goods.
Bon appetit, mate.