Dinner on the jetty (Stokes Hill wharf). http://www.waterfront.nt.gov.au/darwin-waterfront-precinct/stokes-hill-wharf/
X 2............start with lunch and laze your way into dinner with a short stroll around the harbour, wave pool, and oil tunnels in between
or stroll around town on the hill area overlooking the wharf area if you into buildings (Parliament, Government House, Courts, and a few old buildings) -
There's also-
Territory wildlife park -
http://www.territorywildlifepark.com.au/ about 1/2 day to stroll around.
Fanny Bay Goal -
http://www.magnt.net.au/fannie-bay-gaol Couple of hours in the morning on your way to the wharf precinct.
Mindil Beach Sunset Markets -
http://www.mindil.com.au/to-do only on certain days, lots of hippy s#!t stalls, but a really good selection of food stalls. Grab a feed, sit on the beach for sunset, then stroll around the stalls for a looksee......there's usually some live music around the place after dark (I spent about 2 hours watching George Kamikawa busk one time)
The Museum has an interesting Cyclone Tracey display -
https://www.tripadvisor.com.au/Attraction_Review-g255066-d274070-Reviews-Museum_Art_Gallery_of_the_Northern_Territory-Darwin_Top_End_Northern_Territory.html#REVIEWS and if that sort of thing floats your boat, the war museum could be worth a look -
http://www.darwinmilitarymuseum.com.au/ (how many people know that the Japs dropped more bombs on Darwin than Pearl Harbour ?)
Fair way from town, probably a full day trip, but a jumping croc tour from Adelaide river is worth the effort (time it for a lunch stop at the Humpty Doo pub, it's an ordinary pub, but great food) -
http://www.adelaiderivercruises.com.au/tour/jumping-crocodile-cruise/Hard to say what you SHOULD see, as everyone has different tastes, but the above is what we've done in out 2 visits to Darwin (each visit was only for a couple of days). Litchfield is close, and definitely somewhere that you want to go.......... but I recon it's a trip in itself and you want about a week to spend in the area.