Colour coordinate
Designate a colour for each room in the new home, such as yellow for kitchen, orange for dining room etc. Apply coloured stickers on the box near the box number. In your new home. Put a matching sticker on the door to each room. The movers will know where to put everything when they arrive at the destination. It's also helpful to post a big sign on the wall in the room where you want boxes stacked, ("Boxes here Please") to keep them out of furniture and traffic area's.

If you must clean your old place after moving out, put together a kit of basic cleaning supplies and rags. Clean anything possible ahead of time (the inside of Kitchen cupboards, the oven, windows,etc) and if possible, Vacuum each room as movers empty it.

Keep important papers with you
Your list of "important" papers might include: Birth certificates,school records,movers estimates,new job contracts, utility company numbers,recent bank records, current bills,phone lists, closing papers, realtor info, maps and more. Don't leave these with the mover. Keep them with you!