I might be embarking on a similar trip next year—same route. I'm not familiar with the American climate; What would be the ideal time of year to do such a journey? February/March to July/August?
I did the northern route, NH to OR, July-Sept. It was getting a bit chilly towards the end, but the scary weather is storms you hit in the midwest in the open plains. Those tend to be worst in the late summer, I think. It's pretty terrifying on a bike in the middle of nowhere, with plains as far as the eye can see, when the horizon ahead turns black and you just watch the lightning bear down on you. People out there are really nice, though, and would often pull over and offer shelter.