I've never done a winter road trip, but I've always wanted to. You'll have to have the right vehicle if you're headed to certain parts of the country in the winter. For example, many parts of I-70 and I-90 close during bad weather. The Rockies can get particularly bad. Snow tires are a must for these parts!
In terms of things to bring, I'd pack plenty of dry food, water, and a small suitcase. Doing your laundry often is sometimes easier than carrying extra weight in your suitcase.
You can also check out my hub on things to bring on a road trip: