When I/we lived in Kenya for a while we went pretty well to all the game reserves, etc. and the one that we all liked best was Samburu, which just happens to be pretty much the most difficult to get to. Even if you are not going far from Nairobi, you can see more or less all types of game in Nairobi Park literally on the edge of the town (one edge of it, the fenced one, runs along part of the airport road) with the exception of elephants. The Sheldrake Trust have (or at least had) an elephant orphanage on the edge of Nairobi Park, well worth visiting (properly, i.e. when they take the calves out for a walk).