I looked up this model of camera to see the spec. Here's a review:
http://www.ukdigitalcamerareviews.co.uk/panasonic_lumix_dmc-fx12_reviews.htm
First, you can change the light sensitivity setting. If you are going clubbing or want to take shots in low light you can crank up the ISO as high as 1250. This will greatly increase your ability to take shots in poor light, but is likely to lead to some "noise" in the image (speckling visible on areas of the same colour). On the other hand, if you use this setting and flash, you will greatly increase the flash range. For ordinary daylight shooting you should set it at 100 or 200 to minimise noise. The settings available to you are 100/200/400/800/1250 (and an auto setting) where 200 is twice as sensitive as 100, 400 is twice as sensitive as 200 etc, so 1250 is 3.5 "stops" more sensitive than 100. Your user manual will tell you how to change the ISO setting. I see it also has a "party" setting which may do all of this automatically for you.
The reason your shots aren't so blurry is that the camera includes a "vibration reduction" system. In the review it says:
"The Panasonic FX12 has 7.2 effective megapixels and a MEGA Optical Image Stabiliser (OIS) which compensates for handshake blurring by detecting any movement and then moving the lens to compensate, thus producing clear images without any deterioration in quality. The Panasonic Lumix DMC-FX12 also compensates for blurring caused by the subject moving, particularly useful when taking photos of children or sports. The camera can detect the movement and will raise the ISO setting and the shutter speed automatically, resulting in good quality prints."
This means you should be able to get decent shots when ratted and wobbly at a club or party and some good bunny action shots of the mid-binky variety.
However it's your third shot, the cat on the windowsill, which is interesting. It shows that the metering is good enough to give a decent exposure of the fur despite the cat being backlit. Not so long ago compact camera metering would have given you only the silhouette of the cat and a nice shot of the street. You could use a bit of fill-in flash to put catchlights in the cat's eyes and a bit more light into its face but it's a decent shot without.
Do read the manual though. If you end up with the camera on the wrong settings your shots may not come out as you expect.