Your Plan-
I would use NYC as the a fun time for at least 4 nights, its one of the best cities in the world IMO. Then train to Philly.
Then train to Wash DC.
Then after DC, get a rental car, and take oneway to New Orleans.
After New Orleans, fly to Phoenix or Sante Fe, and get a car, and drive to LA.
It is going to be VERY HOT in August in the south, as Hot as it gets, especially out west.
There is nothing to see between Nashville and Santa Fe/Phoenix, except New Orleans.
If nature/hiking/camping is a focus, Smokey Mountian National Park/Blue Ridge Park Way area is the place for this, with the Appalachian Trail a multi day hike....
3 weeks is NOT a lot of time, when deduct out the huge distances you need to travel. if you just go to NYC 2 nights, Philly 1 night, Wash DC 3 nights, thats 1 week, now you have two weeks to go across country...
DC to New Orleans via SMNP/BRP/Nashville/Natchez Trail is another week.
Now you have 1 week, for 3/4's the country.
Have you looked at oneway car rentals? It will be $100 day for a car oneway...plus insurance.
My suggestions-
Culturally, foodie and landscape highlights to focus on would be-
Charleston SC- Culture, History, Low Country cuisine
New Orleans- Culture, History, French Creole cuisine
Asheville NC -is the real Bluegrass heritage, and mother to country music, good hiking and foodie scene too.
San Fran/Sonoma/Wine country- great weather and foodie/wine scene. San Fran fun city for 2 nights.
You need two weeks for the parks out west, best to do a big loop from LA, San Fran or Las Vegas.
Mountains- Colorado, Montana, Wyoming and Utah will be far nicer weather, and cooler, than the southern parks, which will be hottest time of year. Yellowstone NP is the gem of the entire country, need 1 week for that, could fly NO to Denver get a car and drop in Salt Lake City UT or all the way to San Fran.
August is the biggest travel month for US residents, and the key parks are jammed, as well as NYC, Wash DC, and almost everything on the tourist list.