Your first moving box is the first thing that you should pack, and also the first thing that you should unpack. It should contain all of the essentials that you will need for the first few nights in your new home, since it may be a little while before your other belongings get there if you are moving long distance, or until you have completed the full unpacking process, which can be quite long deciding on the size of your home and family. Knowing what some of the most important essentials are can help you avoid headaches on moving day.
Make sure that you bring a towel for each person, as well as soap, shampoo, toothpaste, and of course, toilet paper. You will want to have all of these items handy after a long day of packing and loading boxes to get yourself feeling clean again.
In a very similar train of thought to the above point, you should also remember to pack some cleaning supplies to clean the rooms of your new home. A general purpose cleaner, paper towels or a reusable rag or sponge, and disinfectant wipes are a good way to clean down surfaces before you use them. While it won't be able to fit into a single box, you should also see if you can fit a broom and dustpan into your car, since it's likely that there will be at least some dust accumulation on the floors.
The one thing that you don't want to forget is basic bedding. You don't have to have every single pillow, sheet and blanket stuffed into a single box, but you should make sure to have at the very least one pillow and one blanket for each person (though if it's just you and your spouse, a single blanket is likely fine). In a pinch, consider packing sleeping bags and pillows instead, especially if you're going to be roughing it on the floor before the mattresses and couches arrive.
Especially if you have small children, it is essential for you to pack a number of small snacks that you can hand out throughout the day as people get hungry, since you likely won't have time to sit down for a full meal until you've unloaded the entire moving truck and started to unpack. In a similar line of thinking, make sure that you bring pet food for your pets so that they don't become overly agitated on moving day.
For more help with moving, contact a company like Bekins Van Lines Inc.