Some questions; kind of long

[RichardTaylor]

With hand held low light shooting - the children in the living room..

(1) Know haw to read a histogram (this will tell you if your exposure is technically ok)

(2) Shoot RAW (this will give you maximum control when post processing, including noise reduction and colour balance.

(3) Shoot manual or aperture priority with the lens wide open (smallest f number).
This will gather the maximum amount of light for a given time period.
You may need to dial in some exposure compensation if using aperture priority.

(4) Take what ever shutter speed you can get

(5) If the pics are too dark or blurry due to subject or camera motion then you will need a faster lens.

Or
you can put your camera on a tripod, use a remote release or self timer, shoot at a longer shutter speed and tell everybody to "freeze".

or
use additional lighting (flash?)

(6) If the pics are ok then you may be able to close your aperture down a bit to give you more DOF and/or reduce the ISO (to give you more DOF)

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For sports events in good light (outdoors in daytime) try sports mode.

or (especually if it is a little darker)

Raw.
High ISO
Aperture priority with the lens wide open.
Take what ever shutter speed you can get.

If you find you are getting good images consider dropping the aperture and/or ISO.
If you find the pics are too dark or blurry due to camera or subject movement consider a fast lens.

Keep in mind that when shooting wide open, especially in low light, accurate focussing may become a problem. Some lenses can make this even more difficult.