Your move should occur at a time that is convenient for you. However, You should consider if you want to transfer your children during the school term as well as other factors involving your family. If your move can be rescheduled for another time when the van line agent will be able to accommodate your preferred delivery dates, you may be able to reduce your moving cost.
Many major van lines place several households on the same truck, so your delivery date may depend on the order in which they need to unload the truck. If you need delivery on or by a certain date, be sure to specify this up front. It may affect the cost of your move.
Note that many movers use codes to indicate the condition of a piece and you may need to take a little time to understand their coding. Also note that they may tend to note more wear and tear to your furniture. If you have a particularly valuable or fragile piece of furniture, take pictures or a video of it prior to the move and ask about special crating and handling of it ahead of time.
If you are driving a long distance and need to spend the night somewhere, try to pre-plan where you will stay. Choose somewhere in a decent neighborhood so you can park somewhere safe and invest in a good, strong lock for the truck to deter thieves.