Nairobi N. P.

Nairobi National Park is a treasure which many visitors to Kenya don't discover. It's only a short distance outside the city - right next to the airports - and most of the safari companies do half-day trips, bookable from most city hotels. Alternatively, you could contact one of the safari companies directly (Express Group Travel definitely do this) and book a private trip (take a picnic) - even all day won't let you cover all the park properly. You could hire a car (not recommended on your first trip - a good driver/guide will help you to get the most out of your visit, and will be able to get information about where various animals are.)
In the park, you can see virtually all the animals you will see in the other game parks, except elephant. A river runs through the park giving the opportunity to see hippos, crocs and turtles, and there are some pools. The bird-list for the park is also excellent.
There is a lesser-known migration of wildebeest between Amboseli and Nairobi NP. In July 1999, the wildebeest were in Amboseli when we visited, but by the time we got to Nairobi NP they had arrived there, Our driver-guide said they had just arrived.
Nairobi NP is also a good place to see Black Rhino, as several have been translocated there for protection.