Generally the go-to for photos of your kids is a DSLR. Bulk isn't really a concern since you'll either be at home or somewhere stationary, kid photos can be demanding in terms of available light and time to shoot before a moment passes and shots per second, and you get the best price
erformance ratio there. Canon T3i can be had with kit lens for $560 from Amazon, so within your budget and not $1k.
If you want a more pocketable mirrorless camera then $500 will get you a Sony NEX-3N with kit lens, but the AF performance won't be as fast as a DSLR, and that can be important for getting clutch candid shots. The $650 NEX-5R has phase detect AF, in that case, but it's getting up there in price at that point.