There are various things to do around Darwin. One of its main attractions is its many beaches, with the main ones being Casuarina, Mindil and Vestey. Part of Casuarina is a nude beach so unless you're willing to bare all be careful where you stop.
Darwin is also a great place to explore the Aboriginal culture. Two tours you can join are the White Crane Dreaming Tour to the Kuwuma Djudian peoples land and Peppi Tours to Bathurst Island.
If you head to the north of Darwin you'll come across the Botanic Gardens filled with Aboriginal and tropical plants. If you head further north again you'll reach East Point Reserve where you can walk, ride and feed the wallabies. Closer to the city is Aquascene where you can feed the fish during feeding time. To see more sealife visit the Indo-Pacific Marine and Australian Pearling Exhibition where many creatures and coral can be found in its aquarium, open everyday 9.00am - 6.00pm. They also offer evening shows on Wednesday , Friday and Sunday. Another animal park is Crocodylus Park where you can watch the feedings everyday at 10.00am and 2.00pm, and explore the rest of the park between 9.00am - 5.00pm.
There are also several museums in and around Darwin. The Museum and Art Gallery of Northern Territory is at Fannie Bay, hosting Aboriginal and Asian art as well as a collection of stuffed animals and a maritime exhibition. The museum is open:
Monday - Friday: 9.00am - 5.00pm
Saturday - Sunday: 10.00am - 5.00pm
Further North is the Fannie Bay Gaol Museum (open at the times above) where you can see the old cells and a gallows.