Make an early start with a day trip to the Tiwi Islands. You need to travel with an authorised operator and get a permit, both of which can be arranged through the daily ferry service. Leave at 7.30am and arrive in Nguiu, the main settlement on Bathurst Island, two-and-a-half hours later.

Visit the old missionary church with its ceiling covered in traditional Tiwi art, Tiwi Design (where some of Tiwi's most famous artists work), and the island's small but fascinating museum. There's a fresh barramundi lunch put on before the ferry heads back to Darwin (the return trip can be quite choppy — don't eat too much or you'll lose it all!).

Back on the mainland, head straight out of town on the Arnhem Highway to reach Fogg Dam, an organic farm/bed and breakfast set on the floodplains and run by two long-time Territorians, Heather Boulden and Jeremy Hemphill. Enjoy a sundowner and a swim in the pool before dinner.

It's an early start again the next day — at sunrise, tuck into your picnic breakfast (prepared by Heather with her homemade conserves and fresh produce). Enjoy it as you watch the floodplains come to life. After breakfast head onwards up the Arnhem Highway to Kakadu National Park (about 90km away). At Jabiru, meet up with guides from the Arnhemlander for a trip into Arnhem Land to see the Indigenous rock art at Injalak Hill, Oenpelli.

Injalak is an incredible place; its long rock-ledge caves are filled with art stretching back 40,000 years. Independent travel to it, however, is difficult, as access to Arnhem Land is restricted by permit. Access is also not always guaranteed — roads and entire communities can shut down for ceremonies without warning. The Arnhemlander is a good option because it's owned by the Djabulukgu Aboriginal Association, and as such has access to country that other operators don't: if one area is closed for a ceremony they can take you somewhere else.

After Injalak, take lunch on a canopied boat on Inkiyu billabong, a quaint winding tributary that opens out onto a majestic floodplain with views of the West Kakadu escarpment. Back over the Kakadu border, visit Bowali Visitors Centre — a Glenn Murcutt/Troppo designed building inspired by the long horizontal lines of a rock escarpment. After this you have a choice: escape the afternoon heat with a visit to the centre's impressive museum to learn more about the Bininj, Kakadu's Indigenous people, or see more art at Nourlangie Rock (famous, but nowhere near as impressive as the art at Injalak). Or do both, if you've got time.

As evening approaches, head for Ubirr, meander past more ancient Indigenous paintings and then climb the rock for sunset views over the floodplains made famous by Crocodile Dundee.

Stay the night at Gagudju Lodge, Cooinda. Relax by the pool and enjoy a buffet dinner at the hotel's family-friendly pub or fine dining in the five-star restaurant (be sure to try some freshwater barra). Then head back to Darwin early the next day.