Auckland and Dunedin are at polar opposite ends of New Zealand: Auckland is near the top of the North Island, while Dunedin is near the bottom of the South Island. They're 888 miles (1430 kms) apart, and the easiest way to travel between them is with a direct flight. You can also make it a scenic, multi-day road trip. Here's more on how to get from Auckland to Dunedin.
Duration: 2 hours
Given the distance between Auckland and Dunedin and the fact that they're on different islands, flying is the quickest and most convenient way of traveling between them. Several direct flights make the trip every day.
Duration: 3+ days
Few travelers would want to make the overland journey between Auckland and Dunedin all in one go, but the north-south journey does make for a great New Zealand road trip. There are many possible routes you could take, stopping at points of interest throughout the North and South Islands.
The most direct road journey between Auckland and Dunedin heads south on State Highway (SH) 1 through Hamilton and Matamata (of Hobbiton fame), and skirts Lake Taupo and the Tongariro National Park, before reaching Wellington, at the bottom of the North Island. If driven in one go, it takes about eight hours to get from Auckland to Wellington. The journey to Wellington can also be taken though the west of the North Island, via New Plymouth and Mount Taranaki, east via Rotorua and Napier, or even further east via Tauranga and Gisborne.
Whichever route you take it's necessary to stop in Wellington, probably overnight, because the North and South Islands are separated by the Cook Strait. Take the vehicle ferry to Picton, at the top of the South Island. Ferries leave several times a day from Wellington, from the early hours of the morning through to the late evening. The crossing takes about three hours. It's beautiful on a good day, especially as it sails through the Marlborough Sounds. However, it can also be rough when there's been bad weather, and is sometimes canceled if the sea is too rough.
At Picton, continue driving south. The most direct route to Dunedin follows SH 1 through Blenheim and Kaikoura, along the east coast of the upper South Island, and then through Christchurch, Timaru, and Moeraki before finally reaching Dunedin. Driving directly, the journey from Picton to Dunedin takes about nine hours, but Kaikoura, Christchurch, and Moeraki are good places to break the journey.
A (much) longer way to travel south through the South Island to Dunedin is to head west from Picton to Nelson, then along the West Coast through Westport, Greymouth, Franz Josef, and Fox Glacier, crossing the mountains at the Haast Pass to Queenstown, and then east across to Dunedin.
Duration: 2+ days
The simplest route outlined above (Auckland to Wellington through the Central North Island, and then Picton to Dunedin along the east coast of the South Island) can also be traveled by long-distance bus. These leave Auckland in the early morning or in the evening. You may need to book different legs of the journey separately. As when self-driving, there are many worthwhile stops along the way so it's best not to do the full Auckland to Dunedin bus trip in one go.
By Rail and Road
Duration: 3+ days
The North Island portion of the above journey can also be made by train. The Northern Explorer leaves Auckland three times a week, traveling through the central North Island and arriving in Wellington about 12 hours later. Make your own arrangements to cross to Picton. From Picton, you can take another train to Christchurch (the Coastal Pacific, which runs daily), or rent a car and drive the rest of the way south to Dunedin. Alternatively, you can rent a car in Picton and take one of the two routes outlined above to Dunedin, either going west to Nelson and then south along the West Coast, or going directly south along the east coast to Dunedin.