New Zealand's rates of violent crime are in line with those in other developed countries and you'll almost certainly come across some grisly stories in the media.
Still, as long as you use your common sense, you're unlikely to run into any trouble. Some caution is needed in the seedier quarters of the larger cities where it's unwise to walk alone late at night. One major safety issue is "boy racers" using city and town streets as racetracks for customized cars, leading to bystander fatalities. Although the police do take action, their presence is relatively thin on the ground so be careful when out late in city suburbs.
This sounds like some kind of Pokémon town.