It depends on your budget and who you are with. Primes will be the best quality and depending on where you are, a 300mm is absolute minimum, and 400mm is a decent length for mammals. For birding, you'll need 600mm with a good AF motor. A good safari guide will be able to get you close enough and with the plains, it's easier than paved road game parks to go right up to the animals. However, they cost 2-3x the price of zooms. You can use teleconverters with losing AF with fast primes, whereas with zooms you may go over f/8 and can't AF.
A zoom is the most versatile, especially if you get lucky enough to find a predation. If you get a quality zoom like the Nikon 200-400VR, the image quality is close to primes but not quite the same.
If you don't want to spend too much money on lenses, the Sigma 150-500 is a nice flexible lens. It has image stabilization (OS) and very decent quality. It's got decent to good center resolution with crappy corners wide open, which is pretty much how most of these cheaper zooms will do. I tried a friend's copy a few months back and for $1000, it's good value but obviously not top end stuff, especially on the AF end.
A tripod or monopod (if you are sharing the vehicle with other photogs) with a ball head or even better, a Sidekick or Wimberley head will make it easy to shoot behavior as you can move with the subject without having to shift the bag.
You don't have to buy a sand bag. Since you'll be in Tanzania, just bring a simple freezer ziplock or cloth bag, go to the markets (ask before you take photos) and buy some beans to fill the bag. Less weight to deal with and if you stop in Kenya, get coffee beans from the camp/hotel so you can enjoy the best of both worlds 