For someone who isn't an experienced photographer and doesn't have much inclination to learn a lot before your safari you are probably better off going with a superzoom such as the Canon SX510 you are considering. There are some trade offs with such a camera, but if you are planning on doing most of your shooting during the day you won't notice it as much.

If you want to add more reach to the Rebel T1i the best 'bang for the buck' on the budget end is the EF-S 55-250mm f/4-5.6 IS II. It gives image quality comparable to the more expensive EF 70-300mm f/4-5.6 IS, you just can't use an EF-S lens on a full frame camera (which you don't need to worry about with the Rebel). I've seen some pretty good deals on the 55-250 lately. A good place to watch for alerts on deals from a variety of sellers is the The Digital Picture. If you subscribe to notifications from their facebook page you can keep an eye out for limited time deals.

Update (12/2019): The comparison now is the EF-S 55-250mm f/4-5.6 IS STM vs. the EF 70-300mm f/4-5.6 IS II. Both lenses are a little better than their respective predecessors, but compared to each other they're both very similar in the same way that each of their predecessors were when compared to each other.

For someone who doesn't know how to use the stuff, renting better gear won't make a lot of difference in your results. And the price of rentals can add up in a hurry if you don't live near a good rental house: You have to allow a couple of extra days to make sure they arrive via FedEx/UPS before you leave for your trip and then you probably want to give yourself a cushion on the other end of the rental in case your return flights are delayed for any reason.