Wildlife is one of the most expensive genres of photography because of the long (and expensive) lenses. Great autofocus is also important. The cheapest I'd go is a Lumix G7 (minimum $250) + 14-140mm lens (~$300), both used. And even that isn't going to give you the photos you think it will. More on reddit.com

I shoot birds randomly around my house on these now cheap bodies: Nikon d3100, d300, d90, J1 and Canon 350D and 50D, all with a cheapo no name manual 500mm lens handheld. I was using 300mm for a while but you have to be a lot closer and the critters usually run away before you can get too close. I then tried a teleconverter and it was useless, so I just went with a no name manual 500mm, I wasn't interested in camping with a gilly suit in my backyard just to be able use a 300mm at close range lol.

Here's one with a 300mm from about 40ish feet away. The subject is pretty small. https://i.ibb.co/nRPrzbB/Nikon-D3100-Tamron-70-300-f5-6-1-3200-800-iso-300mm-eq-450mm-Woodpecker-redhead-edited.jpg

Here's one from the 500mm lens, about 40ish feet away again. You can see how much more of the subject fills the frame. https://i.ibb.co/tHYBq45/Nikon-D3100-500mm-f8-1-500sec-iso-1600-500mm-eq750mm-grey-boi-lookin-ass.jpg

Thats why people recommend a 500-800 lens from the start. So you can have it easier by being able to be further away and to be able to fill more of the frame with your subject.

If you need autofocus I would recommend a Tamron 150-600mm, they run for around $400-600 used. Thats the cheapest quality lens I could find that has autofocus. If I took the hobby more seriously I would get that lens but its not what I primarily shoot, so I have no real use for it outside of random home use. Sigma has one too in the same range.

Also I have a handful of bridge cameras I use for street shooting and they are great for that but not for birds. They focus slow and sometimes have a hard time focusing at the longer end, and you cant shoot a fast as a DSLR. You can get away with shooting with a bridge but you will miss a lot of shots just from the wait time of zooming in and out alone, I would just get a DSLR.

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I'd look to see if you could find an Olympus E-M5II and an Oly 75-300mm lens; both might cost used within that budget. Gives you a 600mm equivalent reach, a weather proof body, decent features, great IBIS for handheld shooting, light package. Not the best AF, but the lens is decent and it's some of the better value for money. An E-M10 might also be a good bet for the camera body.

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Wildlife photography is among the most gear-demanding types of photography. Your best bet for your budget would be to get an older DSLR crop sensor body (Canon 7D for example), and then the best telephoto lens you can find for your budget. The 7D is listed on eBay for $250 and is a good starter camera (although dated). You would then need to find a lens for it. You could get a manual telephoto lens (no auto focus) for a couple hundred dollars. Avoid some crazy zoom lens (20-200 for example) as it won't autofocus well, it won't let in a lot of light, and it won't be sharp. I believe this older setup is really the only alternative to your phone as you aren't getting anything made in the last 10 years with a telephoto lens for less than 1K that will be better for birding. More on reddit.com