Haven't had a camera for years. I read great reviews on the Canon T7i EOS DSLR. My wife & I are going on a safari in South Africa and I'm thinking about buying this. I can get it with an 18-55mm lens and 55-250mm lens. Or, should I get the 70-300 lens instead of the 55-250?
Rent something like a 400mm f/5.6 or even 400mm f/2.8.
I read great reviews on the Canon T7i EOS DSLR.
It's a great little camera for its price range, and does a lot of things very well. I'm not sure anyone writing a review of the Rebel T7i/800D would give a glowing report on its appropriateness as a safari camera. It's probably adequate enough to give you many shots you'll like when paired with an appropriate lens, but there are a lot of other bodies out there that will allow you to get even better shots in higher numbers.
For value in a safari package right now, it's hard to beat a Nikon D500 and AF-S FX Nikkor 200-500mm f/5.6E ED VR. The f/5.6 maximum aperture is a bit limiting, but the D500 handles high ISO/low light a bit better than Canon's current APS-C offerings do.¹
If you are considering renting, the more expensive AF-S NIKKOR 200-400mm f/4G ED VR II gives you an extra stop at all focal lengths, gives up 500mm vs. 400mm of focal length, while having better overall image quality.
¹ Just for the record, I'm a Canon shooter and have been since the 1990s. But right now the Nikon D500 is a superior camera to the Canon EOS 7D Mark II, which is the Canon APS-C high performance counterpart.
No, you should get the 55-250.
55-250 is a crop lens. 70-300 is a full frame lens. The full frame lens offers you slight focal length advantage, but I wouldn't say that 300mm will get you shots that 250mm doesn't. 300mm is only 20% more compared to 250mm.
The reason why I'm suggesting 55-250 is that 70-300 is optimized for full frame. For a crop camera, it has worse image quality than 55-250: you'll see image quality benefits only if you use the full image circle. Also, the 70-300 due to its large image circle is significantly heavierweight and significantly more expensive. You're paying for heavy glass you don't really need. In contrast, the 55-250 has an image circle that has been optimized for a crop camera, so there's no unnecessary glass in it.
If you really think you need more focal length, and are willing to pay a heavy monetary sum and a heavier weight for it, consider some 100-400 lens: Canon, Tamron, etc. Unfortunately, they are for full frame so they're needlessly heavy on a crop camera. Also the Canon one probably far exceeds your budget (which you didn't specify, by the way), whereas the Tamron is significantly cheaper but still many times the cost of a 55-250.
I'd go with the 55-250 given your potential camera. It's a very nice lens for a very nice price at a very nice weight. It has its limitations, but they mainly come from its crop sized image circle, and you can't overcome that anyway with a crop camera.
However, that said, if you are taking pictures of very quickly moving animals such as birds, the faster autofocus of 70-300 could have some benefits.