Gee all these differing opinions and techniques...
One thing to consider with kids is that they move around a lot and can't stay put. Also you don't want them killing themselves by knocking a hot light over or blinding themselves with a strobe, etc.
And the other thing is to decide just what you're going for? Irving Penn style portraits with (those so-called cheesy) painter dropcloths? Loose active fun stuff? Something in between?
Also it is good to realize that you can break with photography's convention and instead of making you and your subjects hold still while you "build" the light around them (classic studio situations) you can instead move about and "find" the right light, sometimes guiding your subject to "stand there, turn that way" and explore with your camera.... Most photographers have their heads up their butts about doing that but it works well.
For myself, in most cases, what works is to take a bright, hot Tungsten light and bounce it into the corning of the walls and ceiling so it raises the overall room illumination in a soft yet directional way. Bounce a 300-watt Halogen into the corner, about 7' high in an 8' tall, 12' x 16' room with a white ceiling and light-colored walls and now you can easily shoot ISO 400 at 1/125th and f/2.8 or so.... reasonable compared to no lighting and in general the bounce gives you a nice look that has enough contrast.
You can also add a dialed down on-camera flash -- let it be 2-3 stops down from normal - and it kicks fill into your subject but allows you to capture the environment. Experiment, you can get cool blurry halos around a sharp subject by dragging the shutter while using flash.
Stay aware of window light too! Light the opposite wall in a room and the middle area will probably be a good "shooting zone".
It's a nice fluid way to shoot moving critters like children and drunken naked girls.
A digital camera is really nice for previewing the lighting, even if you end up using film for the real shots.
B&W solves color temperature problems or, with practice, Photoshop can make mixed lighting work.
Actually mixed lighting can usually be tamed to make a single good picture. What is hard to editing to have a consistent fleshtone as people move around in mixed lighting... this continues to madden me and is one of the main reasons why it's nice to replace the tungsten w a big professional strobe and radio sync when you do this professionally. Or resort to B&W (my first choice).
Their is a lot more to it of course but a $25 Home Depot Halogen work light and your camera's on-board flash can do a lot.
When the kids calm down, plop them on a stool in the corner or against a clean wall... bounce the hot light off the opposite wall and have a large, soft source. Or use a window. Or combine your warm Tungsten light and blue window light (as a fill) if shooting B&W.
Remember that the distance the kid is from the background makes a lot of difference, that applies to studio shots or real life.
The umbrellas and photo gadgets can come later... (Lowel is a better built alternative to Smith Victor btw, I really like their light stands and hardware). For strobe, those Elinchrom D-Lites look pretty slick for the money - you can add a Skyport radio remote sender for $100. On camera flash? old school Vivitar 283s for older cameras, get whatever you like from Canon or Nikon for your modern Canon or Nikon. Smaller units are not always a bad thing either.
Finally, start with one light. The second comes in for fill or back up. Add a third when you want to light the background independently from the subject.
This is a great book:
http://www.lowel.com/book.html as it applies to more than just his hot lights, but lighting in general.