In Khartoum/Omdurman you must see the Sufi ritual of drumming and trance dancing, about one hour before sunset and Friday prayer. These rituals take place northwest of the Nile river in Omdurman. Very welcoming, festive atmosphere.
A walk around Tuti Island, situated in the middle of the confluence of the two branches of the Nile, can take about four hours. The less populated northern section is pretty, with its shady lanes, and irrigated fields, and there is a great little coffee stall under a tree on the western side.
The pyramids of Meroe are 2.5 hours north of Khartoum leave early to avoid Khartoum traffic. On the same route visit the sites of Naqa and Musawarat. In theory permits are required before visiting the sites and guidebooks say that you pay beforehand in Khartoum, but as of January 2010 this appears to have changed. Now you pay at each site. Cost is 10 Sudanese Pounds. Naqa and Musawarat are signposted beside the Nile Petrol station about 1hours 15 minutes north of Khartoum and the track is fairly clear but sandy. It is probably good to carry a GPS to avoid getting lost in the bush.
After 4pm take a good coffee at The Egg hotel, with high altitude view over Khartoum, the Nile, and Omdurman, and stay to watch the sunset. Worthwhile.
About 1.5 hours south of Khartoum visit the dam. Just north of the dam downstream the Nile is also very wide; on Friday/Saturday the area is popular is day visitors.
There is good diving near Port Sudan, either on live-aboards or from the new Red Sea Resort north of Port Sudan. Beware the windy season Nov/Dec/Jan/Feb unless you're not prone to seasickness 2.5 hours dingy ride from the coast in rough seas can be testing!.