I did a semester abroad there in undergrad and I stayed in Wellington (the Capitol, at the bottom of the north island.) If you're looking to backpack/hike, then you definitely want to check out the south island. My recommendations are to check to see if you can get a permit to do the Milford track (4-5 days) or the Routeburn track (2-3 days). I did the Routeburn track and it was amazing!
As far as cities go, definitely check out Queenstown and Dunedin. Queenstown is a lil touristy but is absolutely gorgeous. You can bungee jump nearby by at a bridge (forget the name, starts with a K).
Another place to check out is Milford sounds. You can take a boat ride to check them out and I recommend taking the cheapest tour (Juicy was the one I took).
Definitely take a boat across the Cook strait (the water between the north and south island.)
As far as the north island goes, the previous poster covered a good chunk but I would recommend climbing Mount Taranaki since you'll be there in the Summer. Don't worry, you should be able to do it within a day.
If you have any other questions, feel free to PM me.