Team recommendation: looking for a way to take good pictures of remote wildlife

I have been investigating which camera I should use to photograph animals at a distance, like on a safari.

I am very limited in the ability to use a camera & # 39; hardcore & # 39 ;, but I have a very basic understanding of things like aperture, focal length and shutter speed. But I am willing to learn if that is what it takes to get decent quality nature photos.

For close-up images, I think I'm happy to use my phone (Pixel 2) and I have a GoPro for diving. I'm mainly looking for something to take good pictures in the range of ~ 30-80 yards.

I am looking to spend less than $ 600.

So far what I have as options are:
1, a telephoto lens for my phone, would require the use of a tripod (included) but it would only cost around $ 40

2, I have a Rebel T3i given to me; Only body, without lens. I would have to buy a telephoto lens and a battery charger + but I am willing to spend up to $ 500 if necessary, however, I was reviewing some lenses in the $ 250 range, I think "Canon EF-S 55-250mm F4- 5.6 It's STM Lens with UV Protection Filter – 58mm "was one of them. I wonder if the technology in this camera is too outdated to be worth using.

3, I saw an announcement of an "EOS Rebel T6 EF-S 18-55 with EF 75-300mm f / 4-5.6 III reconditioned" which seems very reasonable at $ 280 including shipping

4, I read that a camera without a mirror would be a good option as a cheaper, lighter and easier to use version of what I am looking for, but I can not find a solution that is below $ 1000

I appreciate any advice, thank you!