You can apply for Residence under SMC immediately, as soon as you have a skilled job (at the same time as being on the work visa). It will take longer to process, which is why people get the work visa first to get started on living/working in NZ, while it ticks away taking the time it takes.
People are entitled to use the NZ Health Service under two sets of circumstances that may apply to you. Residents are eligible.
http://www.health.govt.nz/new-zealan...land-residents Also, holders of work visas that will run for 24 months at least are eligible.
http://www.health.govt.nz/new-zealan...-years-or-more Apart from that, if you want private medical insurance (until you get eligible, say), you will have to choose it and take it out yourself.