How much time do you have? And are you flying or driving? You really cannot fathom how utterly massive it is until you've driven across or paid the soaring high airfare. (I'll never understand why it's so bloody expensive to fly around within the USA.)
That said, Texas is IMO the best state, but second (for me) is actually Alabama. It's stunningly beautiful at every turn, and you've got to see Birmingham. It's no wonder they built their city there! Also, there is a place just east of Birmingham called Combat Park where you can DRIVE A TANK! And for a little extra, you can drive it OVER A CAR!
Since you'll already be in the South, I recommend driving across the Mississippi River, just to do it. The best crossing is along highway I-20 between Louisiana and Mississippi. You may not be able to cross from Arkansas into Tennessee on i-40 as the bridge is closed for major repairs and will be for quite some time, but that crossing isn't as amazing anyway, IMO.
Definitely check out New Orleans. It's a completely different place to any other. I don't even know what else to say about New Orleans, because there is so much to see and do (and eat and drink!) there that the experience is utterly unique.
Also it's worth it to spend a couple days in Washington DC. I say that very begrudgingly because I lived there for a few years and I loathe that city, BUT it is 100% worth a visit. There are beautiful war memorials, the White House, Congress, the Library of Congress, the Smithsonians (Art, Air & Space, Natural History, and on and on) -- ALL FREE! Plus Arlington National Cemetery is a truly sacred place; the feeling in the air there is just different. The changing of the guard at the Tomb of the Unknown Soldier is... something to behold.
We currently live in Texas, so if you have any questions or specific recommendations, I'm happy to help! I love this place and have spent many many happy years here, and I've been just about everywhere in this state!
(Ugh, I've nearly written a book! Sorry!)