A few things to help you:
Never drop your shutter below 1/125 if the subject breathes. (newborns)
If it breathes and moves, never below 1/250 (toddlers or active babies)
If it breathes and runs like the wind never below 1/500. The faster it moves the faster your shutter.
The best thing you can do is add light and copious amounts of it.
On board flash: YACK! Fashion yourself a light scoop. There are a ton of tutorials for them on the web. http://www.flickr.com/photos/kingfal/304…
I don't know what your camera is, but don't be so afraid of your ISO-learn to expose for it. If you SLIGHTLY overexpose and then reduce exposure in post processing the noise is hidden or eliminated. If you UNDER expose at all and raise exposure in post processing you are creating more noise.
Use the highlight warnings in your camera to tell you when you are blowing something out and push your exposure to the point JUST before you would have an unacceptable blow out. Then in post processing lower that exposure to perfect. I like to be close to 1 full stop over if I can. I can shoot on my crappy 50D at ISO 12600 and have totally usable images.
This image was shot at 12600 and not nearly as overexposed as I would like it: http://farm6.static.flickr.com/5004/5212…
This is a 100% crop of that image so you can see the noise.
http://farm6.static.flickr.com/5168/5212…
It's just a snapshot and it was DARK in that bar, but you get the idea of what you can do when you properly expose for ISO