Winter Backpacking in New Zealand?

My bf and I are doing a bit of world exploration this summer (June-Aug/Sept) and NZ is one of our longest stops.
We both love camping, biking, hiking, live music (esp. local or folk) and cultural events/sights.

With our interests in mind, is there a specific region that would be able to give us the best NZ experience in 2 weeks? A few days of online research has shown me that travel from north to south island is pretty difficult/expensive, even on train and bus. If we stick with one area, we'd also love to rent out bikes. How feasible are good day trips on bikes?

Keep in mind that we're from Seattle, so rain, coffee,
mountains, volcanoes, moderate cold and ferries don't phase us.

And just to make it more interesting, our goal budget is pretty much $0. Other than splurging for a bungee jump (Just one of those things I must do), we're trying to use only camping, mountain huts and couchsurfing to survive. Any cheap/good food tips?

(PS: Are the Waitomo Caves overrated?)

Thank you!