Intercity took our bikes as long as there was some space in the cargo hold. The rule there is that the driver gets to decide whether your bike goes on (depending on available space), so it helps to remove the pedals and turn your handlebars and cover the oily bits, but this is not always necessary. Most drivers were very friendly and will make sure your bike is properly placed to avoid damage to it.
One time they even went so far as to exchange a small passenger van for a full-size bus for 10 people just so our bikes could get on.
Note that these are not local busses.
Almost all (27) bus routes have a 2 place rack out the front. No bikes inside the bus, but you might get away with a folded one. Central Bus Exchange has special loading doors for bikes. No cost for taking a bike. Minimum wheel size is 16"
There are no passenger trains
Ferry - the Diamond Harbour ferry will take bikes as deck cargo, at discretion of staff. Bikes not to go inside.
Bus - No bikes on any bus unless its a folder. There are bike racks on busses on Waiheke Island, but that's all.
Ferry - free at any time. Different vessels vary, just ask.
Trains will take bikes for free if there's room and at discretion of staff. Don't take a full sized bike at peak time. Folders can go at any time and stored under a seat. Use the middle car of every three, (second and fifth car of a six-carriage train)
Busses for 10 of 29 routes are required to be fitted with bike racks. The others probably have bike racks, but its not contractually required. Aim is for all busses to be bike-racked by July 2016. Minimum wheel size is 20"
No trains or ferries in Dunedin.
Busses - folded bikes only, carried by the passenger, at no additional charge. No bike racks available.
Trains - Bikes travel free, but only folders are allowed at peak time. Stations have bike parking.
Ferry - yes, 6-8 bikes permitted for free per crossing, on a first come first served basis