The best way to explore is by car. Most roads are well maintained and clearly signposted, and a car gives you the freedom to explore attractions such as short walks and waterfalls along side roads.
There are two main ways to drive to the Coromandel from Auckland. First take the Southern Motorway, following signs to Hamilton. Just as you get over the "Bombay Hills," turn left onto State Highway 2 (it's clearly signposted to Coromandel); then take the turnoff to State Highway 25, signposted between the small towns of Maramarua and Mangatarata. Follow the signs to Thames. Allow 1½ to 2½ hours for the 118-km (73-mile) journey. An arguably more scenic—though longer—way is to detour off the Southern Motorway at Manurewa, travel through Clevedon and follow the Seabird Coast, past tiny Kaiaua village, Miranda thermal springs, and alongside the Firth of Thames, home to thousands of seabirds attracted here to the internationally recognized Ramsar wetland of Miranda. From Thames, State Highway 25 is the main loop that travels around the peninsula, and though winding, the road is in good condition.
To reach the Bay of Plenty from Auckland, take the Southern Motorway, following signs to Hamilton. Just past the "Bombay Hills," turn left onto State Highway 2, and stay on Highway 2 all the way to Tauranga, driving through Paeroa, the scenic Karangahake Gorge (stop for walks and photos), Waihi, and Katikati on the way. The driving time between Auckland and Katikati is around 2 hours, 40 minutes. Between Auckland and Tauranga, it's at least 3 hours, 15 minutes.