Ship is docked right in the front of the city so makes it very accessible and easy to get around. There is a free bus that does a city circle. Good place to do some shopping, probably the best place and the cheapest I would say. Nice port area with some great restaurants.
You can walk into the city from the port as it is very close or they had a bus which I think was $9 for the day. We visited the Te Papa Museum which is very good, then walked to catch the cable car up to the top of the botanical gardens, great view from here and then a pretty walk down to the city again, it doesn't take a long time and is very easy, we then visited Parliament House and did a tour which was interesting and finished off with a bit of shopping.
We had arranged to hire a car here which worked very well, They did have a shuttle bus at a cost to go into Dunedin because the ship actually arrives in Port Chalmers which would be 20 minutes away from the city centre. Not much to do in Port Chalmers I would think. We travelled down the Otago Peninsula which is beautiful, then had time to do the city sights and then head north for awhile before heading back to the ship so it was good value to hire the car, even better if there were four of you.