My advice is to give up believing you can get an zoomed-in action shot of animals 100 yards away with point and shoot cameras. If something is far away, getting a nice up close shot of it is almost impossible without heavy bulky equipment. Putting large cheap zoom lenses on cheap point and shoot cameras will make every over picture you take with that camera substandard.

My advice is to get a nice small (minimal zoom) point and shoot camera from a reputable company (Canon, Nikon and a few others). Do not focus on zoom range or megapixels; in my experience these play no role in producing nice images. I consider point and shoot cameras pretty disposable. I buy a cheap Canon (~$100-150) every year or two.

In terms of technique. Holding the camera steady is most important. Secondly don't expect more from you equipment than it can do (telephoto action shots, low light objects in the distance). Thirdly spend some time thinking about how you want to frame the photo. People don't seem to spend a lot of time framing there photos. (it is a pet peeve of mine when people in photos are cut-off below the knees.)