Most New Zealand vegetable gardens run into a fairly predictable cast of pests over a season. Aphids and slugs and snails get talked about the most, and rightly so, they're genuinely the most common complaints from home gardeners, and this site has dedicated in-depth articles on dealing with both. This overview is deliberately broader: a roundup of the other pests worth knowing how to spot, rather than a repeat of that detail.
Whitefly is one of the most widespread. It's a tiny, pale, moth-like insect that clusters on the undersides of leaves and scatters into the air in a small cloud when disturbed, which is the easiest way to tell it apart from aphids, which barely move when a leaf is brushed. Tomatoes, cucumber, zucchini, pumpkin and capsicum are all commonly affected. Like aphids, whitefly feeds by sucking sap and can build up quickly on soft new growth, so checking leaf undersides regularly on these crops catches an infestation early, before numbers get out of hand.
Whitefly is one of the most widespread. It's a tiny, pale, moth-like insect that clusters on the undersides of leaves and scatters into the air in a small cloud when disturbed
Caterpillars are the other pest almost every home gardener eventually meets, cabbage white butterfly caterpillars especially, which target brassicas and are the main reason netting young transplants is such standard advice. This site's dedicated article on spotting garden "worms" covers how to tell a caterpillar apart from a harmless earthworm, plus the other caterpillar species to watch for, in more detail; the short version here is that checking leaf undersides for caterpillars and the small egg clusters that precede them catches damage before it gets serious.
Thrips are less obvious than either of the above, since the damage they cause, fine silvery streaking on leaves, is easier to notice than the tiny insects themselves. They turn up on alliums like garlic and spring onion as well as on capsicum, and tend to build up in warm, dry conditions. They're harder to spot early than aphids or whitefly simply because of their size, so it often pays to recognise the leaf damage pattern rather than trying to spot the insect itself.
Leafroller caterpillars and guava moth are worth knowing about too, even though they're more commonly associated with fruiting trees than vegetable beds specifically; again, the spotting-worms article has the identification detail. The practical point for a vegetable garden is that both are worth watching for if your beds share space with fruit trees, which is common in a lot of New Zealand backyards, since guava moth especially, now established from Northland down through Waikato, can turn its attention to fruiting vegetable crops nearby as readily as to a plum or feijoa tree.
None of these pests are usually cause for panic on their own. Regular checking of leaf undersides, netting susceptible crops like brassicas early, and acting while numbers are still low are the same basic habits that work across nearly all of them, whitefly, caterpillars, thrips, leafroller and guava moth alike. For aphids and slugs and snails specifically, this site's dedicated articles go into the detail on identification and non-chemical control that this overview has deliberately left out.