I agree with those people who suggest geting out of Auckland. If you want to stay within reasonable distance of Auckland, the Coromandel Peninsula and north up to the Bay of Islands and Cape Reinga are definitely worth it, as already mentioned. You could also go to Rotorua, which offers Maori cultural experiences and hot pools and geysers, or to Waitomo Caves which have glow worms and beautiful limestone formations.

As far as getting around, buses are more common than trains. If you decide to drive, you should not have to worry about ice in the North Island unless you choose to go into mountain areas in the central North Island.