Broad generalization here but... stuff I have noticed that the average New Zealander often gets for a quick take out - Butter chicken (mild chicken curry), fish & chips, meat pies and sausage rolls. I would say don't waste your time on any of these.
At home, Kiwis also eat a lot of sausages. These are the kind bought at the grocery store and are also not worth eating. Especially avoid the little red ones nicknamed "cheerios". Good sausages can be found in NZ, but not at the grocery store. Watties beans on toast also seems to be a local comfort food/snack but I can't stomach it.
If you have money to burn, crayfish is okay. But it's really expensive and tastes like lobster only a little tougher.
Things worth trying:
No matter what restaurant you end up at - kumara and snapper are worth getting. Kumara (like a sweet potato) comes in a variety of types - gold, red skinned, orange, etc. - and to me they are all really good. Green lipped mussels are also good and anything with passionfruit in it.
Burger Fuel is a popular local burger chain. Definitely try this place if you like burgers. And don't be afraid to try the Kiwi burger with beet (locally called beetroot).
If you find a place that is known for having good whitebait fritters - try them. They were a little scary to me at first (the fish looks like small worms) but they're not that bad.
Vlad, I dont think there is a ' locals favorite dish', just like there wont be in any other city in the world. Auckland is a very cosmopolitan city and most things you enjoy eating, we in the whole of New Zealand do too!! Actually, you can be sure the last things any Kiwi wants to eat out would be Roast Lamb followed by Pavlova. Not that anything is wrong with those dishes, but they are kinda...common. To us at least.;)
NZ is such a young country that we have never developed regional cuisines. And that is a shame really given regional produce.
You will find most International cuisines represented at Auckland restaurants. For some excellent reviews, go visit www.cuisine.co.nz. And if you get a chance, visit Vinnies restaurant. Wonderful.
Enjoy your stay!