Khartoum is the capital city of Sudan. It sits at the confluence of the Blue and the White Nile, whose branches divide the city into three parts - Khartoum, Omdurman and Bahri (Khartoum North).
North towards Atbara, Kerima, Wadi Halfa
Take any of the minibuses that head north towards El Droshab in Bahri. From here it is very easy to flag down rides heading out of the city. Don't use a sign. You will come to a police check-point as you head out, and if you are polite and friendly, they will normally find you a vehicle to hop in.
South towards Wad Madani, Gedaref, Kassala
Take a bus from Souq El-Arabi to Soba at the southern outskirts of the city for 2 SDG. Get off at the last station, which is already on the road to Wad Madani. Continue walking a bit along the road and start hitching from there. You should get a ride within a few minutes.
Hitchhiking within the city
It is certainly possible to hitchhike within the city, even at night. The main streets tend to be very long and it is quite easy to get a ride in the direction you are going. Khartoum is a large and drawn-out city and even though public transport is decent and affordable it can at times be sporadic, especially at night.
Places to avoid
Khartoum is infact a very safe city and the locals will often go out of their way to help you. It would be best to avoid being near military and government buildings, and take no pictures of these either.
Accommodation and Sleep
There are places to camp down by the river, and a very cool place would be near the confluence of the two Niles. You can take a bus out here. People will be curious, but rarely give you problems. Camping should be possible on Tuti Island, which is located between the White and the Blue Nile and accessibly by bridge. Most locals there are extremely friendly and getting invited to stay at somebody's house is fairly common.
You might also try and find a roof to sleep on. Often apartment buildings are open to go into, and there is always a nice spot on a roof in the cool of the night.
Khartoum also has a great Couchsurfing community to boot.
Other useful info
If you are heading South towards Ethiopia, many buses depart to Gedaref. These are cheap, have AC and make stops for food.