I am a photographer (proof)
Reading a few replies here compelled me to create an account and reply.
I suggest you buy an SLR. Not a 'high end compact' or 4/3rds camera. Get an SLR. The 'bottom end' canon/nikon SLRs are incredible these days.
I didn't want to get technical but it's important you understand the technicalities of why it is happening......
Your pictures are blurry because you are trying to do something wich exceeds the capabilities of your camera. It is trying to compensate for this by opening the shutter for longer, which means that the shake of your hand, or the movement of your child, becomes a factor. Unfortunately SLRs still have these constraints: While an SLR's low light capabilities are going to be a lot better than your current camrea, it is still going to get to the point where it is restricted in the same way your current camera is: by your lens's maximum aperture (which is the hole, or 'aperture', inside the lens can only open so wide, letting in only so much amount of light). This is where, as people are mentioning, buying a 'fast' 'prime' lens comes in, because the standard (…cheap) lenses you get with SLRs in fact have similar maximum aperture constraints as your current point&shoot camera - they're great for bright daylight use but not great for low light.
Two things to note: A prime lens means that it is only one focal length - you can't zoom in or out... you have to physically move. Most prime lenses are also quite 'zoomed', (not very wide angled), unless you spend incredible amounts of money you can't really get around that one. ‘Fast’ is referring to the size of the aperture, a ‘fast’ lens means that the aperture hole is large – letting in a lot more light.
If you buy a Canon SLR I’d recommend this:
If you buy a Nikon SLR I’d recommend you buy this:
They are both prime (the canon is zoomed to 50mm, and the Nikon to 35mm – which gives you a wider, more natural angle)
They are both fast – f1.8 lets in a hell of a lot more light than the standard lens that will come with your SLR & the lens that is on your current camera.
All SLRs in the price range you are looking at perform exactly the same way as your current point and shoot camera does – fully automatically. You just point and click. The only difference being, you’re doing it with a much higher quality camera. You don’t have to be scared just because they are ‘professional’ cameras.
Can I suggest (once you have your SLR, otherwise it’ll not make much sense) you watch this video – it’s just basic rundown on proper focusing technique – something which will also really help you get better pictures of people who don’t like to sit still!
Also p.s. buy new. Don't buy second hand. For all you know it has had 90,000 clicks (shutter mechanisms don't last forever). Also a £300 new SLR will take better pictures than a £1,300 SLR from three years ago. Will also be warrantied. Just buy new.