Best cameras for animal photography?

Thought I'd ask around to see what people recommend. I'm somewhat of a beginner to photography but I know that I'd like to focus on animals, particularly my trio of fancy rats and while the camera I currently have is pretty decent, I think I'd like to upgrade. I want a bit more quality and more speed. I've tried taking pics of the rats on 'pet mode' where it takes a series of photos in rapid succession but the 'rapid' part is a bit lacking, I need just a little more speed to catch my rats in the midst of their goofy expressions.

The camera I currently have is a Nikon Coolpix. Given good lighting with a still target it can take a decent picture but again, I'd like to upgrade to something that can take a faster shot of a higher quality, at a relatively close distance since these are my pets that I'll be focusing on who are pretty small.

Money-wise I'd like to shoot for under $1,000 unless you can convince me otherwise. If you're recommending a camera you own then please consider posting or linking to some unedited pictures you've taken with it.

Thank you!