Blurry pictures in low/inside light

[Sweet72947]

So I have a little Nikon Coolpix P&S, 12 Megapixels, 3.6X zoom. I have difficulty getting a decent picture when I am taking them inside, even if the light is good inside. Most of my pictures are outside in the sunlight since I am taking pics of dogs, but when I want to take pics of cats in the cattery, I struggle with getting anything not blurry! Also, if I take pics when its nearing dusk and the sun is no longer super bright in the sky, my outside pics turn out blurry!

I usually have it set on portrait (if taking action shots, I set it to "sports"), and adjust the flash settings to my needs. I have tried it set on "night" and "dusk" and any setting that looks like it might help, but nothing works.

Anybody have any suggestions?