Khartoum is a sprawling tri-city metropolis — Khartoum proper, Omdurman across the White Nile, and Khartoum North (Bahri) across the Blue Nile — and you will not see much of it on foot. Most travellers rely on yellow taxis or shared minibus amjaads; agree fares in advance with taxis, as meters are rarely used.
Shared minibuses (ras) link the three cities and most neighbourhoods. They run set routes and charge a few Sudanese pounds, but the system is not signposted in English, so a friendly local or your hotel can help you understand which one to flag. Tuk-tuks (rickshaws) work the residential areas and are cheaper for short trips.
The Sufi tomb visits in Omdurman, the souks, and the river crossings are all easy half-day or full-day excursions. Walking the downtown grid is fine during the cooler months, but in the heat take a taxi even for distances that look short on the map — the sun does not forgive.