If you are taking a long cross-country road trip, you will likely not have much room to spare in your car once you have packed all your luggage, snacks and people. Even if space is at a premium, do not forget to keep a box of emergency supplies somewhere in the car (ideally, not at the bottom of the trunk).
In a sturdy tote bag or box, you should have a good first aid kit, a flashlight, jumper cables (and know how to use them!), a blanket, work gloves, some basic tools such as screwdrivers and pliers, some food and a jug of water. If you do any driving at night, keep some road flares in your kit as well.
Hopefully, you have some kind of road service coverage (like AAA). Keep your membership number and their phone number handy as well.
Nothing will spoil a vacation more than nasty car trouble, so be prepared.