Most cities are linked by air routes and bus routes. If you get the chance don't miss crossing Cook Strait by ferry either as the Marborough sounds are beautiful, especially in the warmer months.
Trains which still run are Wellington to Auckland. Picton (Marlborough Sounds) to Christchurch and as mentioned the trans Alpine Christchurch to Greymouth. Fares on the trans Alpine are not cheap as it is a purely tourism route.
Rental cars or rental motor homes are popular but not cheap. If you intend spending any real time here possibly buying a banger for
NZ$1000 could be better.
