Because you live in New Zealand, I'm not sure of all the different exchanges you can use where you're at. I would recommend Coinbase, but the closest country to you they work in is Australia.
Looks like there's NZBCX, which is a New Zealand Bitcoin Exchange. I would recommend buying as much Bitcoin as you feel comfortable with.
From there, I would create an account on Poloniex, or any other exchange you prefer, and trade/convert your Bitcoin to a different cryptocurrency.
The first step is buying Bitcoin with your fiat.
The second step is sending your newly acquired Bitcoin to an exchange for trading.
The third step is trading your Bitcoin for a cryptocurrency you want to invest in.
The final step is storing your long term positions in a wallet.
I would probably recommend converting your Bitcoin (BTC) to Ether (ETH), and making Ethereum your long term investment. Alternatively, there's nothing wrong with just buying up as much Bitcoin (BTC) as you can afford and hold it for the long term.
If you have any questions, please post up and we can explain or help along the way.