Spatial data infrastructures are not well developed in East Africa, and there are few places to go to download good data - and the data you do find on an aggregated site is usually of dubious quality. Metadata about origin, validity and scale are usually missing. Things are also changing very rapidly, and you have to be really certain about the provenance of data before you use it.
User's groups at country and regional level, though, can be really helpful. Try joining and posting to the Tanzania GIS User's Group (TZGISUG) - archives are here, main website is www.TZGISUG.org. They have members from all over East Africa (Kenya and Uganda don't seem to have national user's groups, or at least don't advertise very well) and "where do I find data set X" questions tend to get answered pretty quickly.
Another good resource is the OSGeo Africa list, which focuses mainly on South African (and southern) datasets and mapping issues but also has a wide range of members.
Addendum: the site linked in another answer has an excellent warning in red text which is particularly apt:
Beware: The data linked to below may be inaccurate, incomplete, or
just plain wrong. As always, critically examine the data you are
using, look at what organisation produced it and what agenda they may
have, and beware that there are disputes over some of the data
(particularly country boundaries).
Even some of the data available on our TZGISUG site deserves that warning - administrative boundaries in particular are out of date.
The lack of national SDIs mean that much of the data you mention in the edit to your question is locked up in individual institutions, often with no (or vague) data release policy, and versions of that data tend to float around with increasingly muddied pedigrees. Going to the source, ideally people working in the relevant field, is vital.