I make a six hour trip eight times a year for the last five years due to college. I definitely recommend putting your computer on its side, so the mobo faces up. I haven't had any huge issues after the move (making the trip right now actually lol). I like to put blankets around the case. Most issues you may encounter after the trip canbe fixed by simply reseating your equipment to make sure it's in tight.
The biggest risk is damage to your GPU and CPU fan. I regret not removing both during trips, my CPU fan must have broken during one of my trips and its hard to tell if the GPU was damaged at all but it's an easy removal.
As for your monitor, I like to keep the styrofoam that comes with the original packaging and box for trips. Lay them with the backside facing down.