In Auckland, Cosset
is definitely worth the trip out to the 'burbs. The hotcakes are amazing, and it routinely gets excellent reviews from non-veg coffee snobs. They're super friendly as well, and will know what's on in terms of music/events etc.
In the CBD, try Revel on Karangahape Road for coffee - they always have some vegan baked goods and meals as well. Revive on Lorne St is a vegetarian (but not all-vegan) salad bar.
Piha is the best beach around Auckland - I'm pretty sure it has a camp site.
My favourite camping spot in the North Island is at Lake Waikaremoana but it's hard to get to without a car (the last time I went it was anyway, but that was about 8 years ago). The Coromandel is really beautiful as well and much more accessible - you can hike (we say tramp in NZ) The Pinnacles while you're there. Most hikes/tramps are on land managed by the Department of Conservation who also manage a lot of camp sites, and huts on longer tracks. Their website is generally pretty useful: http://www.doc.govt.nz/parks-and-recreation/places-to-stay/
I don't know if you'll be in Wellington but in terms of food Cafe Deluxe (under The Embassy Theatre), Midnight Espresso and Fidel's on Cuba Street are good for vegan options. As are the kebab shops which abound all over the city. Maranui Surf Cafe in Lyall Bay and Pranah (vegetarian, lots of vegan options) in Newtown are also good for eating, but frequently busy.
The San Francisco Bath House, Bodega Bar and The Adelaide usually have something going on (music-wise) most nights of the week.
I'm in London right now so a little out of the loop with what's going on, and don't have first hand knowledge of hostels so can't help there. If you have any other specific questions, let me know. Have a good trip!