We are also on this cruise. We did the first 14 day leg AUK - SYD 2 years ago and now we're going back for the full AUK-SING run this time.
I too try to avoid ship's tours because we are not so much into doing group activities that require everyone to hussle along at the same speed. For the most part, we like to rent a car in many of the ports where we want to experience life away from the port. If you do this, book well ahead on the internet.
Some activities we did on the last cruise that you may like:
Tauranga: rented car and drove to Te Puke - Kiwi farm. Drove to Okere falls and McLaren Falls. On a second visit there we played golf at Tauranga Golf Course
Wellington: Te Papa Museum, Wellington Cable Car to Botanical Gdns. BTW, we were also here twice and we now refer to it as WWW - Windy Wet Wellington:-)
Picton: Lovely little town where we just stayed around the marinia having a few cold ones. I then climbed up to the top of a small mountain to take pictures of the coastal area. There are a lot of see and swim with the dolphin and fishing tours available there too.
Christchurch: Bought the triple pass to do the tram ride, the punting on the Avon, and then out to Gondola ride up the mountain - again, much less money on your own than through the ship. We also did the botanical gdns here too.
Dunedin: Booked the train ride over the internet (1/3 the price) for the ride up the coast to Palmerston. Had a GREAT lunch at a pub called "The Bog"
Burnie: rented a car and drove up the coast to Sister's beach and then to Stanley for outstanding Fish (some type of shark) & Chips
Melbourne: Great public transportation system here. Very helpful Melbourne "Ambassadors" to assist you. Took the tram to Melbourne Zoo - excellent!. Other options are the Queen Victoria markets if you're into that sort of thing, or excellent arts galleries. Don't waste your time going to the Docklands. BTW, Melbourne has some amazing old buildings if your are into that sort of thing.
Sydney: There is lots to see in Circular Quay and Darling Harbour. Of course you should also at least stand on the steps of the Opera House and you may even want to climb the Sydney Harbour Bridge. I didn't climb the bridge because I thought it was a bit pricy and you cannot take photos from up there!
That'll get you started. Contact me privately via email if I can provide any more information.