A few things I learned over the course of about 12 moves.
1. I always set up the house so the packers could easily and quickly pack similar items in the same box(es). Nick nacks all on the dining room table, pictures and photos taken down and in one room, etc. It made unpacking a lot easier.
2. Do a partial DITY, even if you're only taking a couple bags in the car. The paperwork is too easy to skip on the extra cash you are entitled to.
3. Try to do a point to point move. That means, if you can pull it off, have a place to move into at the other end. That way, the driver loads the truck, drives to your new house, and unloads the truck. Most of my damage/loss happened when the truck was offloaded at a warehouse and my stuff crated for temp storage at my new duty station. You won't be able to sightsee driving between duty stations since you will need to beat the driver to your new home. But, if the driver has more than one load for his truck, you may get a little more time if your the first pickup/last drop off.
4. Consider buying the extra insurance that covers up to replacement value, especially if you can't do a point to point move. Otherwise, you will only receive up to depreciated value.
5. Get the packers, loaders, unloaders lunch and drinks. It is hard work and these guys aren't paid that much. They will really appreciate it.
6. Don't hover, but keep an eye on the packers and loaders. Things like CDs, DVDs, video games, etc can easily walk away.
7. This should be a no-brainer, but too often blown off. Don't have small, irreplaceable valuables packed. That heirloom ring or your grandfathers sword. I also made a point of taking important papers/records in my car (bank records, will, insurance, whatever). That way, if the truck burns up on the highway or the crate falls overboard while being shipped overseas, you didn't loose the things that are really hard to live without. We even took wedding album and photo negatives when we had the room.
8. I never had the movers unpack at my new house, except for some of the big items. They won't put it away, even if you know where you want it to go. It only goes on the floor. I could always deal with taking out the paper and boxes later as I unpacked at my own pace. You just need to be sure to unpack within 45 days so you can document any additional damage.
9. If you have the time and don't mind the effort, save your boxes and paper. Put an add in the base paper for them. Lots of people doing DITY moves are looking for boxes.
10. Don't take too long turning in the forms like the 1040R. If it isn't in on time, you are screwed.
11. Take the trash out and wash all the dishes before the packers show up. They will pack the trash, dirty dishes, sleeping dog, anything that is in the house. If there are things you don't want packed, put them in a separate room and clearly label "Do not pack." And then keep reminding them to not pack it.
How is that fraud? You weigh empty and then full.
Not to be too critical, but even if you aren't specifically told about gas and spare tires, just taking an ethical check of yourself should have rung a bell that it isn't the right thing to do.