>>1148467I'm not /p/ so I'm just throwing this out there and it's a little more than your ideal price, but maybe you can get one on sale or used, but I heard good things about the Nikon P900. It has 83x optical zoom and looks like 332x digital which sounds nuts. It's $580 on Amazon but looking at recommended other ones on the page there's a Lumix FZ80 that's 60x optical that's $350. I don't know if either of those are actually good but just check reviews and sample photos. For around $300 you probably won't have too many options, just the leading models of the major brands.
I only have a Canon Powershot 170 IS that's 12x that I got for $140. I like it a lot and it's tiny and fits in my pocket. Usually I can just hold it up to a spotting scope and get a picture that way to make up for the lack of zoom. Pic related is one I took this way, 0 zoom on the camera, 20x on the scope. Quality takes a bit of a hit but it's a compromise. It all just depends on if you want fancy pictures you can frame or just want to ID what you're looking at.