reality is a harsh mistress
Welcome to the RTA Forum!
I'm afraid I see some pretty significant issues that will make your plans very difficult to achieve.
First of all, I suspect you are significantly underestimating the amount of time you need. It takes nearly a week of driving just to cover the miles on a coast to coast trip. You didn't say exactly where you plan to start or where you hope to stop, but with only a week, you really won't have time to make hardly any stops. You really have to ask yourself if sitting in a car for 10 hours a day, for 6 days in a row is really the kind of experience you are really looking for, since that's what doing a coast to coast in a week trip really means.
Second, there is no way around it, doing this as a one way trip will be prohibitively expensive. I'd expect shipping your car back will cost you at least $1000, plus you'd have airfare on top of that. If you rent a car, you should expect to be dealing with at least $25 per day fees for being under 25 (but at least 21) and several hundred dollars for a one way drop fee. In either case, I'd expect you'll be spending at least $2000 total for your group just for the expense of doing this as a one way trip. That doesn't include all the other expenses of food, fuel, lodging, etc that you'll need for the actual roadtrip.
I wish I had better news for you, but in reality, I suspect you'd get far more out of your trip and spend much less money if you abandoned the idea of going coast to coast, and focused instead on having a good time on a more limited roadtrip that fits better with the resources you have available.