The Matakana Coast is lovely. Short distance north of Auckland, and it has rolling hills, orchards and vineyards, winery cafes, the Saturday market in Matakana village, Goat island for snorkeling and the glass bottom boat, Kawau Island boat trip, regional parks, and pretty beaches and bays, tracts of native bush, streams and rivers.
Waiheke Island is also lovely, but you really are best to take a car across on the vehicular ferry to explore. Not a cheap option taking a car to Waiheke. Otherwise you would be parking your existing rental car up for the weekend, and using the island's transport options to explore.
Thames and Coromandel Town along the Firth of Thames are interesting areas. More rugged scenery than the Matakana Coast, but these towns are both historic in terms of former gold-mining and timber milling days. The Karangahake Gorge, and Waihi and Te Aroha are other places to visit and enjoy, for short bush walks and attractive scenery.