Beetles (the order Coleoptera) have hard, movable wing-covers (elytra) instead of front flying wings. They are the largest order of insects.
The true bugs are insects belonging to the Hemiptera order. They have a tube-like mouth with which they pierce plants or animals and suck out liquid.
Cockroaches belong to the Blattodea order. There are more than 30 species of cockroaches in New Zealand, and most of them are native species and generally not found around houses.
Dragonflies and damselflies — the Odonata order.
There are two species of mantids (order Mantodea) in New Zealand, and over 2000 worldwide.