Plenty of good ones already been mentioned - Chaslands highway through the Catlins, Haast Pass and West Coast, Milford Road, Lindis Pass etc...
Highcliff used to be fun but seems a lot bumpier these days and always full of cyclists and tourist buses... come to think of it the buses are probably the reason the road is rougher. For Dunedinites George King Memorial Drive (out near Outram) is really nice.
Moa Flat road (Heriot to Ettrick) is pretty sweet but shhh keep it quiet.
Gravel-wise there is always Taieri Beach, Backline Rd and more others than I can count.
But I think the single most entertaining road I've ever driven, is this one: