The ACA routes are wonderful and carefully researched. One of the chief advantages of a route such as the TransAm is that there are many cyclist services and you will meet a lot of other cyclists. But that can also be a disadvantage if you are wanting to see the U.S. of A. afresh. For two reasons. First, you are just another of a long stream of cyclists that come through each year. And second, there is a tendency for cyclists in a group to have group-centered dynamics.
If you are solo or just with one other cyclist, then you are forced to reach out more. Also, if you strike out to parts unknown, then you are more likely to encounter people who will view your trip with more than a cursory glance.
Then there's the question of quality over quantity. I have usually found that breaking up a long bicycle trip with car detours - bus/train/plane connections - tends to dilute the quality of the overall experience. For me, at least, I get into a simple frame mind when I am touring each and every day. That mindset,, when disrupted, takes a while to get back.
There are four major regions of the United States - Northeast, South, Midwest, and West - and each of these regions has many variations. The Northeast has New England and well as the industrial Mid-Atlantic - highly populated and sometimes tough to negotiate on a bicycle. The South has the Deep South with the strongest African American cultural contribution as well as the Upper South - hill country from Appalachia to the Ozarks. The Midwest has the rich farmlands of the Great Lakes states as well as the wide open expanses of the Great Plains. Finally, the West has not only the California coast but also the Hispanic Southwest and the Rocky Mountain interior.
I agree that east to west is probably best. I would not start before mid April.
My choice would be to start on the Carolina or Georgia coast - in Gullah Geechie country.
Then head north-northeast thru Virginia into the Pennsylvania Dutch part of Pennsylvania.
Wouldn't hurt to head up to the Finger Lakes of New York and catch the Northern Tier,
Then take the Northern Tier across to Midwest to Iowa before striking out across Iowa and Nebraska.
You could then catch the TransAm by heading west from Fort Collins, Colorado thru the Poudre Canyon.
Once you got out to the Pacific, you could either ride down the coast -
Or if you are crunched for time you could hop on Amtrak to get to L.A.
There's also great regional literature that opens up unique perspectives.
Willa Cather's My Antonia is one of the finest works about the Great Plains.
Ivan Doig's This House of Sky is a haunting novel set in the Montana Rockies.
Maya Angelou's I Know Why the Caged Bird Sings addresses rac and poverty in the South.
Sherwood Anderson's Winesburg, Ohio is a classic look at small-town morality in the Midwest.
The list could go on and on - and others may wish to add a few essential titles.
I have found that a book a week has helped me appreciate the world I see.
(Even easier now with a Kindle - Of course, famous titles are available in cheap paperback.)
If you are planning on starting on April 1, you will likely encounter cool wet weather in the eastern mountains and cool to cold weather in the Rockies if you stick to the the basic TransAm route. June in the Rockies can still be very winter-like. Just FYI.