Just looking at those photos, I can pretty safely say there's nothing wrong with your camera. The shots are 'grainy' (i.e the red shirt isnt blurred, it's just grainy), and that's due to it being a low light shot, even with flash you can't expect a compact to use lower iso's in that light. The other problem is that the subjects are moving when the photos are being taken. The camera cant give a high enough shutter speed with the built in flash that it has. All in all, these are pretty normal results for a compact camera.
Moral of the story, without a DSLR, that can shoot at higher Iso's, and a much stronger built in flash, you won't be able to 'freeze' movement, and photos will look a lot grainier than they would if you were using a DSLR.
All I can suggest is that you either come to terms with the limitations of a compact camera, or you invest in a DSLR, that can be fitted with an external flash for even more power than a DSLR's flash already has, and performs a hell of a lot better at higher Iso's. If you can return the camera for an exchange and get something like a D40X, paired with the 18-55kit lense, and a 55-200VR zoom lens, then you would be set. Obviously this would cost a bit more than what you spent on the P+S camera. I traded my Canon P+S and used the money on a D80 and 18-70, i haven't looked back since.
I've upped my Iso, now up yours.
Nikon D700, Nikon D80, Micro Nikkor 105 f2.8 AI-S,Nikon 85mm 1.8D, Nikon 50mm 1.8D, Nikon 35mm F2D, Nikon 16-35 f4 VR, Nikon 24-70mm F2.8 AF-S, Nikon 70-200VRII, Nikon 70-300mm VR, 300mm f2.8 AI IF ED, SB-800, SB-900. Noiseware pro,NIK Dfine, Nik Colour Effex, Nik Sharpener Pro