How do I get this quest to start? I've had the suit for a long time, and can't remove it from inventory. I've listened to all the radio plays. Quest never starts. Did I mess it up by taking the suit before the quest started?

I'm guessing you mean the Silver Shroud quest. Have you tried going and talking to Kent in the side room of the Memory Den?

Listen to the silver shroud radio broadcasts in goodneighbor.