Can I hike in New Zealand?

That's a resounding yes although they call it tramping. New Zealand is one of the best hiking destinations in the world and the vast network of trails means you can lose yourself for days in unspoiled wilderness or take many shorter day hikes. The Department of Conservation have a vast system of back country huts for use by trampers and these range from little more than a basic shelter which are free to use through to the Great Walk huts which have more comfort with mattresses, heating, toilets and sometimes hut wardens. Prices for the latter type of hut start from NZ $15 per adult per night. Six month and twelve month passes are available for those who intend to do a lot of tramping. Full details, downloadable brochures and booking forms are available at the Departmemt of Conversation website. 

Otherwise any i-SITE (visitor information centre) will have a wealth of information regarding local walks and there are always plenty of maps and trail routes available, usually for free but sometimes costing a few cents.