For general family photography, a f/2.8 zoom would be a pretty big investment in general. Any particular reason you were looking at that specifically? But to answer your questions, yes, that zoom would work with your 30D, but....
Assuming that your budget would likely be ~$700 if you're looking at the older Sigma:
I would suggest looking at getting a used but good condition 60D (~400) then get a 55-250mm (~150 used) to replace that Quantaray for use in your kids sports. You can also think about upgrading the 18-55mm to the latest version if you want but I don't think the performance gain will be significant enough for that purchase to play out well. If you want a lens for portraits (assumed since you're asking about the 2.8 zoom), I would just grab the 85mm 1.8 (~300) if your budget can stretch.
The 60D would give you 18MP of resolution plus a newer generation of autofocus capability and video capability with a flip out screen (great for taking pictures above the crowd of parents at the games). You'll have better high ISO capability and can still use your 18-55 if you choose to keep that lens. Now, you would need to invest in a new storage card (SD cards) and a new backup battery but those should be small potatoes.
The 55-250mm is one of the better bang for the buck EF-S lenses and arguably the best in class for telephoto (EF-S). It'll get you the photos that you want in good light. For nighttime games, you might have to push the ISO capability on the camera to get non-blurry photos, but it should be able to do something decent.