Saving your own seed sounds like the natural next step once you've grown a good crop, and for some plants it genuinely works well and costs nothing. For others, it's a waste of effort that leaves you disappointed the following season. The difference comes down to how the plant pollinates and whether it's a hybrid.
Self-pollinating vegetables are the good candidates. Tomatoes, beans, peas and lettuce all mostly fertilise themselves rather than relying on insects to carry pollen between different plants, so seed saved from them tends to grow true to the parent plant, provided the parent wasn't itself a hybrid. These are also, conveniently, some of the easiest and most commonly grown crops in a New Zealand home garden, which makes them a sensible place to start if you want to try seed saving for the first time.
Hybrid, or F1, varieties are the main trap to avoid. Seed packets and seedling labels that say F1 are the first-generation offspring of two specific parent plants bred to combine particular traits, and while the plant you buy or grow performs true to that description, seed saved from its fruit doesn't. The next generation reverts toward the original parent varieties in unpredictable ways, so a saved F1 tomato or bean seed can produce a plant that's noticeably worse than the one it came from, in flavour, yield or vigour. Open-pollinated and heirloom varieties don't have this problem, so it's worth checking a seed packet or plant label before deciding a crop is worth saving from.
The next generation reverts toward the original parent varieties in unpredictable ways, so a saved F1 tomato or bean seed can produce a plant that's noticeably worse than the one it came from, in flavour, yield or vigour.
Cross-pollinating crops are the other poor candidate for a beginner, even when they're not hybrids. Pumpkin and zucchini are both cucurbits, and different varieties readily cross with each other via insect pollination, so seed saved from a home garden with more than one squash-family plant growing nearby often produces an unpredictable cross rather than a true copy of either parent. Getting pure seed from these crops on a home scale generally means hand-pollinating flowers yourself and bagging them to keep insects out, which is more effort than most home gardeners want to put in just to save the cost of a seed packet.
The basic technique, once you've picked good self-pollinating candidates, is straightforward. Let seed heads or pods mature and dry as far as possible while still on the plant, rather than picking early, since seed collected too green won't have finished developing and stores poorly. Once collected, finish drying it further indoors, in a paper bag in a warm, dry spot out of direct sun, until it's properly crunchy rather than just surface-dry. Damp seed stored too soon is prone to rotting or growing mould in storage instead of keeping through to next season.
Storage matters as much as the drying. Keep finished seed in a cool, dark, dry place, an airtight glass jar or metal tin rather than a paper envelope alone or a plastic bag, since paper doesn't fully exclude moisture and rats and other pests can chew through plastic. Label everything clearly with the variety and the date collected: seed left unlabelled for even one season is easy to lose track of, and viability drops the longer it sits in storage regardless of how well it's kept.
Keep your expectations realistic. Even with a good self-pollinating, open-pollinated crop, saved seed from a home garden won't be as reliably uniform as a commercially produced packet, and the odd off-type plant turning up isn't a sign you've done anything wrong. Treat it as a way to grow more of what already did well for you at no extra cost, not as a full replacement for buying seed, especially for crops like pumpkin and zucchini where cross-pollination makes the results genuinely unpredictable.