Visiting Kakadu is extremely dependent on the weather. This part of Australia has two distinct seasons - wet and dry - and much of the national park is inaccessible during the wet season. As the dry begins, different parts of the park open up at different times, as the unsealed roads become passable and areas are cleared of crocodiles. I mention all of this at the start just as advice for anyone thinking of visiting - make sure you think about when you're going and what you'll be able to see. (For the record, wet season can be quite incredible because the waterfalls are in full flow, but you may want to see them by helicopter, rather than road, for instance.)
I visited in dry season, when most things were accessible, and spent three nights in Kakadu. I felt like that was just the right amount of time - anything shorter and I would've left feeling like I had missed things. I certainly could've stayed longer and visited even more parts of the park, but I think I covered all of the main elements.
The landscapes change a lot throughout the park because it's so big (about the size of Slovenia). There are areas of dramatic sandstone escarpments, expansive wetlands full of birds, as well as lots of woodland. You'll get more out of the visit if you're able to experience this variety, and some highlights are stone country at Burrungkuy (Nourlangie), wetlands at Yellow Water, and the swimming hole at Maguk. If you have time, other natural highlights are Anbangbang Billabong and Gunlom.
A big part of the park is the Indigenous culture, and there are usually events at the visitors centre where you can learn about different aspects. But the rock art sites are the real highlights. The easiest to access is Burrungkuy (Nourlangie), but nearby Nanguluwurr is also really impressive without the crowds. Ubirr is also a very significant site.
I think Kakadu is one of the best Australian World Heritage Sites. It's not just visually dramatic, but it has a huge variety of things to see. The blend of natural and cultural is really vivid and one of the things that makes it so significant. Although it is possible to do a day trip from Darwin, I would highly recommend against it because, with all the travel time, you really won't see much. Regardless of whether you'll drive yourself or take a tour, try to stay a couple of nights to appreciate it properly.