If you are short on time (or money) and do not wish to do a full safari, then Nairobi has its own small version right by the actual city.
You can see a large number of animals, though some said this was more like a UK safari Park in scale than a real wilderness such as the Massai Mara.
The Nairobi Safari Walk is inside a park located 7 km from Nairobi city. It's a nice recreation facility with a protected game reserve. You can see the most common animals in Africa, but they are behind glass or fences.
I'm not really can suggest the Safari Walk at the Nairobi National Park. If you have not much time you should do it. If you have more time you should go in with car for the other parts of the park. I was visibly disappointed of the Safari Walk. You must have good luck to see all the animals which you see at the signs. And if you see it, you will see mostly only one of them. One thing was very intersting they had 2 albinos of Zebras.
But for this I can suggest the small Restaurant near the Safari Walk.
If I ever get the chance to go there, I'll make a game drive there. Is already in the plan of my list, and I'm sure this is a good idea. You should do it also.
One of the most amazing things on our walking safari was coming across this giraffe. We rounded a corner, and there was this enormous animal, grazing just a few yards from us. Really magical.