If you and the other party are in agreement, you can arrange and pay for a test yourselves. There is no need in that situation to make an application to the Family Court. You should however consider taking legal advice.
Typically after the application is made to the Family Court, the Court issues a recommendation that DNA testing is done. It is then up to the mother and the putative (possible) father to arrange for a sample of their DNA to be collected.
Generally, the mother takes the child to an approved laboratory, with photo ID for the mother, and a printed photograph of each of them. A blood sample is taken from the adult, and a cheek swab from the child. The photos are kept on file as proof of the person from whom the sample was taken. The father will also attend the laboratory (not at the same time), with photo ID and a photo of himself.
In Court proceedings, identification is a very important part of the process, so that the testing laboratory can conclusively report whether or not the alleged father is indeed the father. There is a clear "chain of custody" so there can be no doubt whose samples were taken.
The only laboratory who does this Court-approved testing in New Zealand is DNA Diagnostics. Their website has useful information about the testing process, and cost.
In Dunedin, the laboratory that takes the samples is Southern Community Laboratories at 95 Hanover Street. You will need to make an appointment first.
If a mother or father is eligible for legal aid, the costs of the test are usually covered by the legal aid grant. It may of course have to be paid back, but that is at an affordable rate.
If the test is not required for Court, then you could look into the various online companies who offer DNA testing. This could be appropriate if the mother and putative father get on well, can cooperate to send the samples away and can be trusted to accept the results once they are in. But if there are any doubts about the other parent’s ability to maturely and calmly accept the results of the test, a Family Court application is likely to be the best option. Contact a specialist family lawyer to discuss your options and the best way forward.