Just did Kenya/Tanzania with a Nikon D3100 body and a 55-300mm Nikkor Lens. Even that was too much at many points, since two of the parks we went to allowed off road driving and we got REALLY close to the animals where I had to resort to a point-n-shoot and even an iPhone 4S (which took some pretty stunning pictures, believe it or not). I never bothered changing to an 18-55 lens during the trip - there was rarely enough time, and you risk dust in your camera.
I did handheld shooting the entire time (no bean bags) and rarely had issues. The driver should stop the car and turns off the engine to reduce vibration...
Most of the newer Land Rovers also had chargers built-into the car, so no worries about running low on juice (carry your charger with you). I did entire 10+ hour outings, on a single, fresh battery. Most high-end safari camps will have bean bags in the car as well.
I shot mostly in aperture priority mode f/5.6 (except during sunset, or into the evening when the camera chose slow shutter speeds creating blurry photos).
Definitely shoot in RAW format so you don't have to worry to much about exposure settings... During the daytime, I usually shot at ISO200 or 400, amping that up as the night time came around.