Hi. I have a Nikon Coolpix P510 -- I bought it awhile ago, maybe a year or so, and it was in the $300 range, but maybe just above the $300 line. It is a very good camera, takes amazing shots (even on auto), has a viewfinder and a 40x optical zoom. I haven't dedicated the time to figure out all the features it has, but it has a very decent auto feature and plenty of manual options if you get experimental.
The things I was looking for with this camera was that it had a rechargeable battery and that it used standard SD card, both of which the P510 have.
While it is true that you "get what you pay for", there's nothing wrong with a $300 camera, unless you intend to lug around three bags full of equipment for the camera, including interchangeable lenses, etc. Unless you're looking to take "professional" pictures where you'll want different types of lenses to manipulate exposure, etc, you're good in the $300 range, honestly.
The only thing I would mention is that if you have tremors in your hand, you may want to reconsider shooting with long range zoom anyway, unless you're using tripods. The higher the zoom, the more sensitive the camera is to movement, even if you've got it up to your eye -- at a very high zoom, I brace my arm somewhere if I don't have a tripod in order to avoid blurriness from movement.
The P510 does have a FANTASTIC zoom, though. I've taken detailed shots of figures on the windows at EPCOT Center's Italy pavilion -- caught the details of a dragonfly posing on a flower from far enough away that I didn't frighten it - the details of a butterfly and hummingbird up in a tree. All on auto settings, now that I think about it.
I think we're not allowed to have links to outside websites on here, so check out trusted reviews website or cnet website for the review of the P510 if you want more information about it.
Anyway, hope you find something good!