Are you getting a green box on the LCD or a beep before pressing the shutter the rest of the way? That is what tells you the camera has focused properly. One thing I found is that you need to be very careful not to move the camera once you press the shutter all the way. Try mounting on a tripod and use the 2 second delay. Indoor shots (even with a lot of light) tend to have slower shutter speeds, so even a slight movement at the time of shutter activation can cause an out of focus image.
Since you are using the camera in Auto and P modes, there is the possibility that the camera is using a very high ISO and the image is actually more noisy than out of focus. The photo would appear very soft with a lack of nice detail. I would use the Tv mode and use a shutter speed of at least 1/125 second and ISO of about 400 or 800 and see if the image improves. If it doesn't, then use the flash as well to see if that helps at all. I also have a PowerShot SX series camera and had issues with indoor shots until I started using more manual modes. These cameras are really good stepping stones for learning photography without breaking the bank. Once you learn the manual modes, you should see an improvement in the quality of the images you take. Plus the photos will come out the way you like.
Steve M.
