Getting out to at least one of the Gulf Islands during your stay in Auckland is highly recommended. Grab your hat, water bottle and lunch and head on out to Rangitoto to climb around on the dormant volcano for a day. If you're brave you can even seek out the lava caves.
Waiheke Island has beautiful beaches, a small settlement and wineries. You can do a tour or just hire a car. Being small and not very busy, this might be a good place to get your driving experience up if you are not used to a right hand drive.
Tiritiri Matangi is a wildlife sanctuary and you'll need to plan ahead for a day trip as it can book up during the summer when visitor numbers are limited.
There are also more remote possibilities but transport or limited. Ask at the ferry building or search online if you are looking for helicopter flights.