First decide if you want to see the Great Smokeys and New Orleans or Savannah and Florida.
Let me say first- I hate Greyhound bus travel. If I don't need to be somewhere ASAP, I rely on Amtrak.
Start either in Washington DC or Charlotte, NC and ride the Crescent to Atlanta. Stay a day and ride the Crescent to New Orleans. Stay 2 days. Ride the City of New Orleans to Memphis.
See Graceland and any other sites you want to see. Ride the CONO to Chicago. See the city. Ride the Capitol Limited to Washington DC. Spend some time there before heading home.
The 15 day USA Railpass is the best deal you will find.
If you don't drive, it will be very hard to see the Great Smokeys. (Maybe a tour bus company).
Riding Amtrak from Washington DC to Savannah and Florida is another easy option.

tom offers excellent suggestions. His train route would be much preferable to Greyhound travel. If there are places you want to see that aren't in his list, join a tour group for a few days (or longer).

Right. If you went to all those places on a bus it would take you 3 months. It's a large area, and the bus stops at every crossroad and driveway -augh.

could she get off the train after she leaves DC to see the Smokeys,maybe join a tour etc,same would be true for trip to NO.Just came back fromSavannah, thats a secial place

PS just looked at the route map DC tp Atl--it stops in Charlottesvvill, a terrific spot to see the smokeys and Monticello,Jeffersons home.Also stops in Toccoa Ga,a great hiking area as well. Might work for you,if you can hook up with local tours

A link to possible tours/buses in the Great Smokies:
http://www.aboutthegreatsmokies.com/tour-guides.html
Best info I could find.

I've taken the Greyhound bus in some parts of the South before, specifically from Atlanta to Montgomery (round trip), Atlanta to Savannah, Memphis to Little Rock (round trip), Cincinnati to Nashville, Memphis to Baton Rouge, and Baton Rouge to New Orleans. I definitely don't see this as being dangerous to do, though there's definitely no luxury feel to this mode of travel. The advantage to a bus over a train is that buses go to more cities, are usually cheaper, and are often more timely -- the one notable lateness exception for me was Atlanta to Savannah, which is apparently chronically very late.

Trains are generally more pleasant than buses but also often cost more. Many train routes also have serious delays -- the one exception in this region I experienced was the one from Memphis to Jackson MS, which was very timely. Amtrak's website lists the on-time rates for all its routes.

Of the cities you listed, I've been to all but Asheville and Knoxville.

2-3 days is good for Savannah, 3-4 days for Charleston. Both are lovely cities with lots of history/architecture and easy to explore without a car, as most of the sights are within walking distance.

You could easily spend 8-10 days in New Orleans if you're thorough -- again lots of history/architecture, terrific food, and some very good museums, including one of the South's best art museums -- and easy to manage without a car.

Atlanta is not an especially attractive city, but you could easily spend as many as 5 days sightseeing there, and public transportation is decent. Some good museums and historical attractions here.

Much of what's interesting in Memphis can be seen in 3-4 days. There are some very good music-oriented attractions and smaller museums, and the city is a BBQ mecca. I found a lot of it walkable (or reached via shuttles) when staying downtown. A car can be useful if you're exploring further afield, though I was able to use public transportation at times. Public transportation gets a bad rap here, but I found it useful -- expect a lot of very poor folks and not much in the way of commuters on the buses.

You can do fine in Nashville without a car if you stay downtown -- the bus hub is about a block from the state capitol building. There are some good historic buildings here, plus a lot of (mostly country) music oriented attractions. I found 3-4 days to be plenty.

As far as I know, the closest approach you'll make to Great Smoky Mountains National Park via bus is Asheville. There might be tours available from there, but I'm not sure about that. You need a car to explore many national parks, and for that matter to even get to most of them.

Where you stay in most of these cities matters a good bit. I'd do the needed research before booking anything, both to make sure you have a good location and a decent place to stay.

I don't think I understand how this will happen once she is "somwhere". How will she get around the GSMNP. The US is pretty notorious for its cities not having good public transportation.
Bus travel is truly gross. I don't want to discourage, but reality needs to be addressed.
Charleston is a compact and VERY interesting city. It does have a little trolley.
New Orleans would also be a good stop. Nashville, since someone has given info on possible public transportation.
and yanno, since this person is obviously not american, our nation's capital is a true world class city, with more than enough to do for days and days. Americans need to learn that DC is not just for dysfunctional musings by elected officials!!