I'd recommend I-80 to I-74 at Davenport, IA. It adds less than 10 miles vs going all the way to Chicago and I-65, but should see much less traffic. The route Dave mentioned using US-36 across Missouri would be ok, but while it is mostly a 4 lane road, its not quite freeway standards and I think sticking to the Interstates will be a little easier for you.

The route to Fayetteville is a bit more complicated, as there's no direct interstate route, but I'd be looking at I-70 to I-75 at Dayton, then US-35 to Charleston WV, then I-77/I-74 to Winston-Salem, and US-421/NC-87 into Fayetteville.

The most concerning thing to me about your plans is that you plan to use both a hitch-haul carrier AND a car top carrier. If you are planning to use both, then it sounds to me like you're going to have your car pretty severely overloaded. That can start to make handling the car very unstable and even dangerous. Especially considering you're dealing with a compact car and an inexperienced driver, I'd recommend you skip the car top carrier, and if you can't fit everything inside the car or on the hitch hauler, then ship the rest of it.