It's a long time since i was in Darwin, but i would recommend (as most did before) Litchfield National park (Waterfalls), Katherine Gorge (Can go up the river by canoe/speed boat and/or make some walks arround) and of course Kakadu National park (JimJim Falls, Aboriginal Paintings, Crocodiles and bird viewing etc..). Maybe you can visit Edith Falls (Small lake with some waterfalls) on the way as well. As i know currently is dry season so the water could be lesser. The main attraction in the north is the nature. Darwin itself is a relative quiet place. Visit the tourist website of the Northern Territories (en.travelnt.com). That will give you a good idea what to expect and what to look for.
Enjoy your trip.