Kia Ora MAKS2,
For accommodation in the Bay of Islands, you need to book soon. Paihia becomes very busy over the NZ summer. As will the bay cruises. There is also accommodation available in Kerikeri (service centre for the BOI). Russell across the bay has more limited accommodation, as does Opua.
For a scenic self drive, be sure to stop overnight close to Auckland airport. Arriving into Auckland and planning to drive the same day out of the super-city is risky due to travel fatigue and unfamiliarity with our roads and traffic flows.
You could drive north on SH1, but turnoff at Kaiwaka for Mangawhai and Mangawhai Heads. Continue to Langs Beach and Waipu Cove after lunch. Both are pretty beaches with views over sea to the Hen and Chicks islands. Whangarei city has a good spread of accommodation. The Town Basin is good for a late afternoon stroll and dining options.
Returning south, head to the Hokianga Harbour, continue south through the Waipoua Forest and Matakohe (kauri) museum. Stop overnight at Warkworth, and SH1/Northern Expressway back to Auckland.
Rotorua and Taupo have thermal attractions, and lake cruises. If you decide to head south, the same advice to stop for your first night in Auckland. Collect a rental car on day 2. Hobbiton, Waitomo Caves and Hamilton Gardens are highlights for tourists visiting the thermal region on the way south, or driving north, to/from Rotorua and Taupo.