Start with each piece as a different object, that is each piece which you want to fall into place at a different time. In edit mode press P and choose loose parts or selection to break larger objects into smaller pieces.
Go to the end of the animation and insert a keyframe for the location of each piece. A to select all, then I->Location. now go back to the beginning of the animation and move everything up so that it is out of the camera view. Insert another keyframe, I->Location.
Now that everything is falling at the same time, go into the dopesheet and stagger the start and end times for each piece.
While it might have a learning curve to start, you may also want to consider using animation nodes.