If you have a choice I suggest you pick our late summer /early fall ( Aug-early October) seasons, it tends to be much drier in most parts of the US rather than May/June which many years are quite wet.
Temperatures will vary, depending on the area you can easily see a lot of variation both seasonally and in the course of a day depending on the area. Let me explain, for example in September I did a long ride from New Jersey to Georgia and back, 750 miles each way. I saw 41F and 102F that day, it was in the 50s when I left at 2am, cooler crossing the mountains of Virginia (Blue Ridge parkway area), and 102F in Georgia around 2PM, and back to 50s again by the time I got back home around 1am. You need to carry a variety of clothing, I went from wishing I had worn my heated gear to melting wearing my thinnest mesh jacket.
I rode from Pennsylvania to Wyoming a couple of times, both times in July, on a trip like that expect anything from hot dry weather under a blazing sun to pouring rain, to hail, to lightning, to a dust storm, the back to sunshine and a gorgeous rainbow.. oh and that was all in one afternoon in Rapid City, SD, near Sturgis. Not trying to scare you, I love long rides, just be aware our weather is often changeable.
Unlike most here I'm an interstate rider, you'll usually find me tearing up the superslab at ludicrous speed to get where I want to go, then I spend my time on the back roads and on excursions to see the sights. The Blue Ridge parkway was probably the longest non interstate ride I ever did, it's awesome, ride every inch of it if you can, but be aware it's SLOW, you need a couple of days.
Keep in touch, sounds like your planning a cool trip!