My take (from chasing down a 3 and 1 year old nephew and niece respectively):
- Relatively fast lens (e.g. nifty-fifty, or the f1.4). Usually stopped down a little (f2-f2.8) for a little bit more DOF and quality
- Servo AF
- High speed continuous (though I rarely fire off a burst... but just in case)
- Accept high ISO for fast shutter speeds. Kids move. You're not just fighting motion blur and a frequent lack of light; shallow DOF due to a large aperture means the subject can move out of the plane of focus pretty quickly too. A bit of noise on a portrait shot isn't anywhere near as objectionable as a shot that's blurred or out of focus.
- When the light levels drop too far for available light, use a decent flashgun, bounced off a wall or ceiling (aim the gun where you'd want your 'virtual light source' to be). Manual, 1/200s, ISO 100, and chimp the ISO, flash power and aperture to control light levels
- Boosting the shadows in Lightroom seems to improve many of my shots, as well as a little lightening of the eyes with a targeted adjustment
most important thing - play! Get the kids comfortable with the camera and you. There's nothing more obvious, and worse, than a shot of a kid that's obviously nervous or self concious because some idiot is pointing this overly large shiny lens at them and trying to goad them into grinning.
I've found it's been useful to get the 3 year old to 'help' me take photos of his little sister & mum - liveview and a cable release (with me holding the camera) means can he engage, and understand it's just fun. I also spend a bit of time playing with both before I try to dive in with a camera - don't make a fuss about taking the shots - that just distracts them into unnatural behaviour.
An example (EXIF not present on that small version, but it's a 7D with a 50mm f1.4 at f2.2, 1/60s, ISO320):