What does it mean when you accelerate from a dead stop and the car makes a grinding noise for a couple of seconds?

It sounds to me like you might have a brake shoe dragging. Try coasting to a stop and then accelerate and see if the noise is still there. The noise could be from a front shoe that is hanging up when pressure is applied to stop the car. It could be caused by a bad caliper piston or warped rotors. Kind of a hard problem to diagnose but sounds like the most probable cause is in the brake system. Hope this helps