Sony Alpha SLT-A35: Low-budget action shooter
Unlike many cameras that have scaled back burst-shooting capabilities, the SLT-A35 delivers fast continuous shooting with a buffer deep enough to make it useful and a viewfinder that makes it possible.
Sony's SLT series of digital cameras is a bit hard to categorize. The company considers them single-lens reflex (SLR) cameras, but they lack the reflexes; they have fixed, transparent mirrors that don't flip up during shooting the way a dSLR's does. That means they also lack a through-the-lens-type optical viewfinder (OVF), instead incorporating an electronic viewfinder (EVF) like the ones you get in megazoom cameras with insanely long lenses.
What they do share with SLRs is the phase-detection autofocus-system technology, which tends to be faster--especially for burst shooting--than the contrast AF systems used by other types of cameras. And the company's latest, entry-level model in its SLT lineup, the Alpha SLT-A35, outperforms its inexpensive dSLR competitors for burst shooting, making it one of the few cameras under $1,000 I'd recommend for people who want to shoot kids' sports.
It's more than just speed alone that makes a camera good for continuous shooting. The viewfinder affects how well you can keep the subject in the frame. Most EVFs black out or don't refresh quickly enough so that you can keep the subject in the viewfinder if it's moving laterally; really, you spend a lot of time simply hoping you got the shot you thought you did. So I don't usually recommend cameras with EVFs for this type of shooting. But while the SLT's EVF wasn't quite as seamless as a dSLRs OVF--OVFs blackout from the mirror flip is so short you barely perceive it--it was good enough that I could pan for short bursts.
The camera's not perfect; it has other issues that might bother you. But if you're looking for a (relatively) inexpensive camera that can keep up with your kids, it's worth checking out.