Amboseli is one of the smaller reserves (national parks) in Kenya. It lies on the Kenyan Tanzanian border directly south of Nairobi. It's about a 4 hr drive out of the capital.
I have a personal affinity for this park. It is the area where Cynthia Moss has been doing her elephant research for over 30 years. I have read many books about the place, visited the area and feel that I know it fairly well.
There is so much wildlife in Amboseli, all packed into this "smallish" reserve, that there is always something to see right on your doorstep. The elephants are especially revered here. The herds were almost decimated by ivory poachers back in the 1970's and 80's, such that the "ellies" became very shy and reclusive. Their numbers were very near the non-recoverable point when the Kenyan Wildlife Service (under Richard Leakey) finally put an end to the poaching. Since then, the elephant population has grown and is now back to a very sustainable size. They are everywhere - what a magnificent animal.
If one were to visit Kenya for only a short time, a 3-4 day excursion to Amboseli would satisfy the "animal viewing" requirement. Additionally, most of the animals are fairly habituated to humans so you can approach quite close to most of them with impunity (a guide is required for this, just to be on the safe side). They are, after all, still wild animals.