My recommendation is Canon 28mm 1.8. It looks very natural on a 1.6x crop body like yours (equiv. 44.8mm). I've heard nice things about Sigma 30mm 1.4 (equiv. 48mm), but I've also heard many complaining about its poor quality at times.
I do understand the reasons why people are recommending a flash, however if your lens can't focus fast enough, it won't matter because they'll be OOF. For children running around, there's no real solution, other than get a camera that shoots 10fps, shoot and pray. No Canon flash can recycle fast enough at that framerate!
BTW, I've found my 70-200 2.8 IS to be the best lens for shooting kids outdoors, albeit far more than what you're ready to spend. Unfortunately, there aren't too many options indoors.
Regarding your budget, do think about the real cost of ownership. Even if you get a very expensive lens, some lenses hold their value a lot over the years. Unless you aren't going to be using them much, the real cost of lens isn't as high as you may think. Also, kids only grow up once, and there are intangible values on being able to capture the moments. My children will see how I saw them through my viewfinder. I only hope I can preserve those great memories.