Provided you visit Sudan in winter, it's fairly easy to get around by bike. Traffic is minimal, roads are in very good shape and signs are in English.
There are no boat services that can be used within Sudan.
Bus & Minibus
Sudan is undergoing a road-building frenzy and all significant towns northeast of El-Obeid are now linked by excellent paved roads. Fast, comfortable buses have almost totally replaced most of the bokasi (pick-up trucks) that formerly bounced over the desert between big northern towns. There are also plenty of minibuses between the smaller towns; they usually leave when they're full. It's best to buy bus tickets a day in advance.
Car & Motorcycle
Sudan has an excellent road network that links most cities and towns of interest to travellers. Most tour operators in Khartoum can arrange car hire, and all make hiring a driver compulsory with their vehicles. For a 4WD with driver expect to pay around US$110 a day. Fuel is generally extra, but it's very cheap.
Hitching may be a tempting option for travellers on a restricted budget, but traffic is fairly light on Sudan's main roads. If you're determined to hitch, you could try hanging around at one of the main petrol stations in or just out of town. Remember that hitching is always a risky option and best avoided by women.
The only remaining practical passenger service in Sudan is the Khartoum–Wadi Halfa run, but it runs on a monthly basis and there's no official schedule.