Strange sound after starting, maybe gear shift?

There's a quick scrape sound that always happens after starting up for the first time, and after I pick up a little bit of speed. The sound comes from under the car on the driver's side, but it only happens the first time after starting up. Even if I stop at a red light or park the car (but leave it running), the sound won't happen. But if I turn the car off, and back on again, it will make noise after picking up speed.