Everyone in NZ has universal accident cover (called ACC) that provides good quality medical care, and financial support whilst you are recovering from an injury. This covers tourists, people on work visas, everyone, even if you were being reckless. There are levies paid by employers, insurers and in a couple of other ways to fund this cover - you do not do or pay anything to get cover. If you are in NZ, you are covered.
NZ has a public health system which depending on your visa, you may have to pay the full price, or the subsidised price. For primary health care at your GP you will oay about $50 to see your doctor. Hospital treatment etc is usually free, but there are some things which are not subsidised - or your case is outside of the subsidised criteria, so you would need to pay full price. In general urgent stuff is handled pretty well. Non urgent stuff tends to have long waiting lists. Southern cross medical insurance provides cover for scans and chronic care, and is provided by some employers. Health insurance in NZ does not provide acute / emergency cover.
Life insurance cover is limited to residents and citizens. Some other insurances such as redundancy cover are only available as a bolt on to life insurance. Coverage seems better than in the UK for things like suicide, however, as you get older and have built up more retirement savings, life insurance may become more expensive and less critical, should things go south.