My lingering memory of Darwin is barramundi... you basically can't get any other fish besides barra, though it's usually fried and dripping with old smelly oil - sigh. If it doesn't say grilled on the menu don't bother.
My last trip to Darwin was very Italian. I suggest Chianti's in the city, at Cullen Bay I suggest the pizza at Seadogs, and if you want to buy your way into the record books there's a $300 foccacia at Il Piatto at SkyCity which was slightly beyond my means :-)
If you're crowd friendly the Mindil markets have some surprisingly good food. If you're self catering I rate the gourmet deli in Parap, they have very decent bread and a good range of cheese and smallgoods (including, last I was there, jamon iberico, bone in).