indoor - How can I take pictures of active children with a DSLR in low light?

I am not an expert. I am shooting images of children mostly in indoor conditions when the lighting is not perfect and keep getting the message subject is too dark. Also the pictures come as very dark.

I am shooting with a Nikon D5100 but that shouldn't make much difference.

I noticed speed of less than 1/100 sec makes the images blurry. So I can't go any lower than that on the speed.

Also I noticed the ISO of more than 400 makes the images grainy.

Another problem is that most children don't react well to flash light and they close their eyes. My desired mode of taking pictures is continuous. I take 3-4 continuous shot so I can later pick the best posture. So even because of that flash is not appropriate since I can't capture continuous shots.

These are my findings so far, and I was thinking to try a bright light source, but I am not sure whether children reaction to the light source would be also that great.

Any suggestion how I can have a better experience for indoor shooting. I know I am asking for a bit much here but maybe there are some tips I can follow to enhance the experience. Afterall I noticed with my iphone camera can take bright pictures (and not blurry). So I really like to do the same with my DSLR.

Update: I am shooting in S mode, so aperture is automatically set by the camera, and when I receive the warning subject is too dark, the aperture is at its widest set by the camera.

I noticed my widest aperture is 5.6. Maybe as some people suggested changing the lens to one with a wider aperture can help.

Answer

The answer @eftpotrm gave is pretty comprehensive, but let me highlight the single piece of advice that is by far the most likely to give you the desired results:

Get a lens with a large maximum aperture, like f/1.8 !!

The smaller the number, the better, but f/1.8 is the best that's typically available at a reasonable price. It's going to be a prime lens (i.e. no zoom).

A larger aperture will allow the lens to gather more light, and when shooting in low light situations with quick movement, that is what it's all about.