For me the Northern Territory is one of the most beautiful places in Australia. It's probably because I love the outback. There is a vast space and no many people. So little that it is not considering as a State, but Territory.
Alice Springs is a major town in the Red Centre. The name originated from the soil colour. For me it is amazing how beautiful and wild can be here. Alice Springs itself depends on tourism, but there is little to see in the town. I think worth to see can be Flying Doctor centre - institution which allows medical assistance in remote and isolated parts of the territory. Besides, Alice Springs is rather a starting point to surrounded attractions such as Uluru (460 km), Kings Canyon (320 km), Macdonnell Range (110 km). Distances are given along paved road from Alice Springs to the main park centre.
Getting to Uluru, Kata Tjuta and Kings Canyon you can rent a car or take a package tour. Personally, I got there by hitchhiking on my second day in Australia. It was definitely cheaper version of the following prices. The most popular trip takes three days and two nights, costs from 230 AUD. You sleep under the stars, and because the temperature can drop below zero even in the summer, sleeping bag can be hired for duration of the trip for around 10-15 AUD. As usually we can find cheaper flights from Ayers Rock airport (from Alice Springs is more expensive), so we can arrange a trip that after sightseeing the bus will take us to the Ayers Rock airport and the rest of the group will continue to Alice Springs (or we can start at Ayers Rock).
Uluru (Ayers Rock) is a symbol of Australia. This red rock is lying in the heart of the country. Often mistakenly reported to be the largest monolith in the world, but no matter, it’s huge and impressive. You should be at different times of the day in Uluru NP, especially at sunrise and sunset. Then you can really see the changing colours of burgundy, red, orange, etc. Before the sun reaches its zenith I recommend to walk around the rock (10.6 km, 3.5 hours). Many of the passing places are sacred for Aborigines, no photography is allowed. Hike, however, is sensational. If it’s no raining, which does not happen too often here, you can climb to the top of Uluru (867 m). however Aborigines are asking not to do it, because this monolith is their holy place. Unfortunately, the situation of Aborigines in the country is not easy, and for average Sydneysiders they are associated with crime, drunkenness and begging. This social problem is very complicated and so far hasn’t been resolved. To understand a bit the problem of "stolen generations" I refer to the movie "Rabbit-proof fence”, details in the movies. Aborigines were finally apologized by Australian Government in 2008.
Driving 50 km further from Uluru you can find Kata Tjuta (Olgas Mt.). This is also an interesting rock formation with walking trails. We can choose Gorge Walp track, 2.6 km, 1 hour return. More interesting version is 7 km trail called Valley of the Winds. We should spend on the loop about three hours. Along the way there is tap water. In the summer, this trail may be closed around 10 am due to heat.
Admission fee to the both parks costs 25 AUD AND is valid for three days with the possibility of re-enter the park. Campsite in Yulara Resort starts from 36 AUD per two-person tent. Double room is an expense comfort - from 200 AUD. I hung my hammock near the resort for free. Although it was a summer night, it was very handy to me to have a good sleeping bag, beanie and gloves.
Further north we come to Kings Canyon (Wattarka Park). You can walk around the top rim (6km, 3-4 hours). You’ll be rewarded with the views of red sandstone from 300m height. The place called The Garden of Eden is constantly filled waterhole, where you can swim. For lovers of the longer journey, there is also a 2-day, 22 km Giles track.
MacDonnell mountain range offers countless hiking trails. The longest (223 km) and most popular one is Larapinta Trail. It consists 12 sections, so you can just take a part of it. Before leaving, make sure to check the current situation of water supplies and you can arrange the food drops off along the route. MacDonnell is considered as one of the oldest exposed rocks in the world.
Leaving Alice Springs to the north, we’ll pass the Tropic of Capricorn. A little further there Tennant Creek town with nearby gold mines. 105 km from the town, just next to the main road, there are interesting massive granite boulders, called the Devils Marbles. It is worth a stop, walk, take pictures (especially at sunset), and camping is also permitted here (fees apply, but cheap).
In Elsey Park near to Mataranka, we can dip into hot thermal pool, 34 °C. But don’t expect that will be place just for you, it may happen that you will have dozens of people around. Looking for a little privacy, choose a walk to waterfalls, this 8 km track (4 hours return) is no too crowded. Mataranka at summer may be flooded or full of crocodiles from the nearby river.
We won’t miss Katherine town even when we take the famous Ghan train from Adelaide to Darwin. There is in fact mandatory, approximately 4-hour stop. Passengers are offered various kinds of activity in the nearby
Nitmiluk Park - with its famous Katherine Gorge. We can take kayaking for 58 AUD, boat cruise for 85 AUD (kayaks and boats do not operate in the rainy season from October to April), helicopter scenic flight for 205 AUD, or simply enjoy a gorge lookout (minibus transfer fro 25 AUD). If you have more time, this park offers numerous trails in length from 2 to 20 km around the gorge. You can also go for a 5-day, 66k km trekking from the Katherine Gorge to the Leliyn waterfalls (Edith Falls). This waterfall can also be reached by car from the other side. We must be careful with swimming, of course park rangers inform about the situation, but always look at crocodile notice board. If any, it should be just freshwater crocodile, not so dangerous, but anyway it may be shock for swimmer.
Approaching Darwin we can see the green Litchfield Park. Litchfield Park. There are a lot of waterholes with waterfalls, termitaries, green vegetation etc. Numerous walking trails, mostly short ones to the lookouts. Lots of places to swim, but not everywhere is safe - just read the signs before entering the water. The park is nice, but it seems to me that it should be visited first in the region, as compared to Kakadu 
it’s not as impressive. It is better to graduate attraction, rather than decrease their level. Day trip from Darwin will cost around 100 AUD. If you are looking for more privacy spots, go to those most popular ones early morning or evening, and leave it when tourist buses has gone.
A little over 100 km away Darwin is located, at the end of National Road No. 1. Nothing special, however, an exception may be Thursday and Sunday sunset market. Darwin is the base for one of the most interesting national parks in Australia - Kakadu.
Remember that during the rainy season (October-April), most roads are impassable. So the best time to visit will be between May and September. In the park best way to go around is by car (admission 25 AUD per person, valid for two weeks), but hitchhiking is still possible. If you don’t have a 4WD, three days tour around the park should be enough. First of all, I recommend cruise on the Yellow Water Billabong, preferably at sunrise or sunset (from 66 to 97 AUD). Then we have a very good opportunity to watch at the flora and fauna, especially for thousands of birds - a paradise for ornithologists. We’ll also encounter the crocodiles, both freshwater (not too dangerous for humans) and huge saltwater.
We can spend the rest of the day strolling along the many hiking trails, where except wildlife we can also see old rock paintings. However I was scared while selecting one of the tracks just before sunrise. Although the park authority said it was safe, then trekking in the darkness a few meters from the river bank where I saw a huge crocodiles the previous day, push my imagination to the limit.
In designated areas park rangers tells stories and facts - ask at the information centre. At sunset I recommend to climb Ubirr rock where there is a lot of old paintings and rock engravings. Hence you can see a wonderful view of the wetlands. After a romantic sunset I suggest fast run to hide inside the car 
from attack of millions bloodthirsty mosquitoes. Park offers plenty camping sites, some are free of charge, others for as little as 5 or 10 AUD per person. If you have a car with four wheel drive, you can go to the famous waterfalls - Jim Jim and Twin Falls. We can also get there by boat (12.50 AUD), but same like by car, only during the dry season. We can swim at Jim Jim watefalls, but you should read the actual warning. At Twin Falls saltwater crocodiles occur regularly, therefore, a bathing is prohibited. The problem is that the waterfalls are not too impressive during the dry season (scenery is great, but not much water in cascades), and are inaccessible in the rainy season. Then we can charter a scenic flight (225 AUD) and have a good look from the bird perspective.
Arnhem Land - Aboriginal land where the white man can get only with special permission (a matter of high fees), and anyway can’t move freely. This is in another word “sanctuary”, a place of free Aborigines, which left there only about 16,000. Arnhem is larger than Portugal.
Like at the Red Centre, if we don’t have own car, we can take advantage of package tours. As the distance from Darwin is large (170 km), at least two days and one night is worth to spend there, but even so it will be in a hurry. Price for two days and one night stay starts from 380 AUD. If we combine Kakadu with Litchfield, I recommend to see Litchfield park as the first attraction.