Thanks for a great question! I think I'll do a blog post specifically about that this week. Some tips off the top of my head...
For free accommodations and work exchanges - couchsurfing, WWOOFing, workaway.info, helpx, staying in a hostel for $
For food - making your own food, in NZ coffee was usually $5 per cup so I carried around a french press and my own coffee grounds, and made sandwiches for the road
For travel - taking buses, hitchhiking (super safe in NZ and pretty common)
NZ also offers 1 year work visas for US, Canadian, and UK citizens
I hope that helps!