I did precisely the same route that
PSC1K mentions when I moved from California to DC six years ago. I wanted my car and I wanted to transport stuff (most of what I owned could fit in my car back then!) so it just made the most sense.
I started in Fresno and made it to Flagstaff the first day (I didn't start early).
Second day I stopped at Meteor Crater, Arizona (they charge way too much to get in, but it's still cool to see a 49,000 year old meteor crater) and I stopped at an Indian trading post in New Mexico. There are gobs of those things lining the freeways. I did get a decent $4 wallet that lasted 3 years. I wound up that night in Shamrock, Texas.
Third day I drove through until Heth, Arkansas (just outside of Earle - no, that shouldn't sound familiar). Very bummed too, because I was only a half an hour or so from Memphis.
The next night I stayed somewhere in Virginia, and made it all the way up to DC the following morning.
Each night I found myself I hotel. The key was that I wanted a hotel that had (or was next to) a bar. Why? Each night I would talk to the locals, and I was somebody different each time.
The most fun was when I had just gotten out of jail. I had paid my debt to society -- and was hoping to start a new life.
Fun, fun, fun. It's a long drive, but well worth doing once.
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More Room Throughout Coach: the Website of Free Miles and Free Markets