It's true if you are indoors and there is not enough light for you to be able to expose with a decent shutter speed, any motion will be captured as blur.
Using built-in flash indoors will almost always render you a shot that is harshly lit and not at all the mood of the moment.
- Sometimes blur adds to the scene!
What I try to do is stop using flash when I'm indoors and start using flash when outdoors. What you say?, that's backwards. Yes I say and it works wonders.
When taking an indoor shot 9 times out 10 the scene is lit by tungsten lights. You MUST alter your camera's white-balance to compensate or your photos will come out orange and dingey. If you know how to manually set your white balance, do it, the payoff is large. If not, set your camera to tungsten. You may also want to boost your ISO if motion blur is a problem. Some cameras are better than others at doing this well without adding a bunch of noise (little colored speckles) all over your photo so your mileage may vary. Then, steady your camera. Lean against the wall, a table, anything, it's better if you can actually hold the camera against the wall while you shoot. This takes practice. Shoot lots of photos. Some will be keepers and some won't be. Snap (I mean take) lots of pictures. Film was like playing poker, I like the odds of digital much better.
Now, how about outside? Did I say to use flash in the bright outdoors earlier? Yes I did. What you are attempting to accomplish by using flash outdoors is to fill dark shadows with some light because often the sun will be so strong as to light your background just fine, but everyone's face will be covered with dark shadows. This is what many snapshots look like, but we're trying to do better than a snapshot right?
One last word. Experiment! and Enjoy!
(ok that was three)