The attached image for example, (f 5.3, 1/80, ISO 800) - no post-processing.
My understanding is that when the subject is moving (like a toddler) you want to have a faster shutter, which makes the image darker, so you turn up the ISO.
In this case, the image is still underexposed, so the ISO should have been turned up even more, but its already really grainy to begin with.
Ok, so the D3000 your dealing with a much older sensor, and older sensors are not nearly as good at noise reduction and shooting at higher ISO's as newer sensor will be - a D3200, for example, is capable of shooting at much higher ISO's than your 3000 before noise becomes noticeable,, for example.
Shutter speed is an important consideration, as is aperture. Faster lenses, say F2.8 or F1.8, will allow in a lot more light than a lens that say goes from 4-5.6. Lower aperture numbers et you shoot at lower ISO's at the same shutter speed because more light is getting in, however it also makes your depth of field more narrow so you need to keep that in mind when choosing your aperture.
Shutter speed of course, the slower the shutter speed the longer it stays open and the more light that gets in, but as you already mentioned the slower the shutter speed the more you run the risk of motion blur and possibly camera shake when your shutter speed drops low enough.
So it's always a trade off - a balancing act. You want to keep your shutter speed high enough but not too high, your aperture low enough but not too low, and hopefuly your ISO down to the pont where the outcome isn't two terribly noisy.
Newer cameras with better sensors can make a difference, as of course can shooting a full frame system as opposed to AFS-C, and faster glass is always a plus as well.
Worst case scenario though, well, a good flash is just tough to beat.