This is an excellent website which seeks to detail holdings of various archival and research institutions in Africa. It occasionally provides contact information, which can be particularly invaluable. For most archives in Africa, as for smaller museums and research holdings in Britain, developing a rapport with an archivist, librarian or scholar is critical.
The Kenya National Archives. This is a very accessible and well-organised archive, though you'll have to be a little patient with document delivery (and occasionally documents will be missing). It's worth getting on good terms with the very helpful staff. Power outages are very infrequent.
Zanzibar Archives Unlike the Tanzania National Archives, research on Zanzibar is much more accessible.
South African National Archives
The National Archives of Zambia, a very well-organised archive, where an extensive digitisation process is proceeding apace, complemented by a nice set of records at the Livingstone Museum.