I don't have any experience with road trips, so if this seems unrealistic let me know.
You kind of answer your own question here, you've never done a roadtrip, yet you think just because you are young, you can do anything.

Being young certainly has some advantages, you'll have more energy and will be able to recover more quickly than someone older. You're also probably more used to be up at odd hours, and all of that helps - for a little while.

I'm still in my 20's and there are trips I did in College, that I probably couldn't do now (or would be smart enough not to try), but that's generally for one shot deals where I gave myself a chance to recover afterwards.

But after a while, the human body starts pushing back. You simply can't sleep as good in a car as you can in a bed. Plus, if you go this route, some one is going to have to be sleeping in the middle of the day. In the best case sincerio, you have to treat this like work, where the people who are not driving or navigating must be sleeping. If for any reason they can't fall asleep or stay asleep, you must abandon the plan.

The big problem with doing a trip with shift driving like this in your case, is that you don't have experience. You don't know your own limits, you don't know how well you'll be able to sleep sitting up, etc. If you don't treat this trip like work, you could very easily kill yourself or someone else on the road.

In any case, I'd think you'd want your Spring Break to be about fun, and the kind of pace and the kind of work you'd have to do to make this trip happen, really wouldn't be much fun. No matter what you do for a roadtrip, it will be a memorable time, but because you are trying to do so much with so little time, I can very easily see your trip turning into an experience that you remember, for all the wrong reasons.