There are lots of options.

You could talk to your bank about a merchant facility or alternatively you could talk to one of the online payment acceptors.

There are often a number of fees associated with this and if you are only receiving occasional credit card transactions you may even be better offering paypal as payment option.

You should also be aware of what happens in the situation when a payment is reversed because the customer has requested this via their Issuer.

It might be worth you visiting a bank and discussing with them and at least you will have their fees/charges/processes to compare with other alternatives.

Paypal, paymate, or your bank. If you need a terminal then you need a provider.

There is a NZ based credit card checkout.
DPS, Like paypal. Probably got huge transfer fees like them..

I use SwipeHQ for taking online payments, and used this for the Geekzone IRL evenings we had last year.

Another good option is KiwiPay.  Both companies have quite stringent sign up processes, so expect to fill in a lot of paperwork, and provide multiple copies of your different IDs.

I would avoid talking to your bank at this stage as you need a reasonable amount of volume before it becomes economical.  As other's have mentioned, Paypal is also a good place to start.