Take a storage box and put a bunch of hanging folders in it. Take a bunch of file folders and put them on one side. Take one of your piles. For each piece of paper, decide if you can throw it out or recycle it. If not, see if it fits in an already-existing folder. If not, take a folder, write something descriptive, and put the piece of paper into it. Place the folder in the storage box. Repeat for all your piles.
Not sure I need a storage box, but I sure need to tackle some of my piles sooner than later. Thanks for the elbow, Johanna!