The backpacker sites are your best guide. NZ has a well established backpacker network. There are accommodation options to suit any budget - backpacker lodges, youth hostels, motor camps with campsites or cabins, motels, pubs, DoC campgrounds etc. Most town have a "domain" where camping is permitted.
Also the are quite a few
Warmshowers hosts in NZ.
You do need to take care camping, there is a serious backlash in NZ against freedom campers who camp, litter and defecate in public camping areas.
You can get a good guide to prices from the many sites like this:
What is the Cost of Backpacking in New Zealand?How Much To Budget For A Month In…New ZealandFor camping food, you can get
Back Country freeze-dried meals in shops almost everywhere in New Zealand, and there are plenty of supermarkets if you are more ambitious about your camp food.
I highly recommend you get the
Wikicamps New Zealand app - it is a fantastic resource.