Looking for a way to take good pictures of wildlife at a distance

I have been doing some research on what camera I should use for travel photography of animals at a distance, such as on safari.

I am very limited on ability to use a ‘hardcore’ camera, but I have a very basic understanding of things like aperture, focal length, and shutter speed. But am willing to learn if that’s what it takes to get decent quality photos of nature.

For close up pictures I think I am content to use my phone (Pixel 2), and I have a GoPro for Scuba diving. I am mostly looking for something to take good pictures at the ~30-80 yard range.

I am looking to spend under $ 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 would only cost about $ 40

2, I do have a Rebel T3i given to me; body only, no lens. I would have to purchase a telephoto lens and a battery + charger but am willing to spend up to $ 500 if I need to, however I was checking out some lenses around the $ 250 range, i think “Canon EF-S 55-250mm F4-5.6 is STM Lens with UV Protection Filter – 58 mm” was one of them. I wonder if the technology of this camera is too outdated to be worth using though.

3, I saw an ad for a “EOS Rebel T6 EF-S 18-55 with EF 75-300mm f/4-5.6 III Refurbished” that seems very reasonable at $ 280 including shipping

4, I read that a mirrorless camera would be a good choice as a cheaper, more lightweight, more user friendly version of what I am looking for, but cannot find a solution that would be under $ 1000

Greatly appreciate any advise, thanks! Alan