Complete your portion of packing job
Complete as much of the packing of the items you're packing yourself as possible by the day before your move. On moving day, take care of any last-minute items first thing in the morning. This serves a dual purpose: only the items being handled by the packers will be left, allowing them to focus solely on what they're going to pack, and your mover will not have to wait while you finish your portion of the packing job.
Remove any obstacles
Before the movers arrive to pack and load, remove any obstacles so they'll have a clear path from the door to the truck. Also check hallways and stairways and eliminate any items in the way. If you'll need to remove a stair railing or other "impediment" to get larger items out of the house (your estimator should have alerted you to this), do so before the movers arrive.
Make a recheck
Take a walk through the house and check closets and utility areas make sure you haven't forgotten to pack anything you were supposed to. Make sure the boxes you packed are clearly labeled and their room location at your destination is clearly marked on the box. Make sure all of your boxes are taped closed.
Taking measures to handle pets
Make accommodations for your pets. You might consider boarding or caging them on moving day, letting them stay with a neighbor or relative during the actual move, or you may be able to better accommodate them by taking them to your destination in advance. You'll want to do what's appropriate in order to keep them out of the way of the movers and help reduce their stress levels and anxiety.
It's good to be on the site
Plan to be around to keep a watchful eye on both the packers and the movers while they're packing and loading your possessions. That way, you'll also be readily available to answer any questions they might have. It doesn't hurt to build rapport with your movers and packers either, as they're more likely to do a good job if they like you.