I will be moving from Seattle to Raleigh at the end of May. Can anyone suggest a route?

I will be driving a 16ft moving truck while towing a car, with my wife and a dog. I would like to cover about 500 miles a day, so I will need to stop at motels along the way that are dog friendly (female yellow lab). I would like to avoid bad weather, and drive the flattest route possible.

Thanks