I am by no means an expert on Amtrak but here's the route I would take:
Starting in Savannah, take the Silver Star or Silver Meteor overnight to Washington D.C., where I would spend time seeing the many tourist attractions. Next, take the Crescent overnight to New Orleans, after which you would take the City of New Orleans to Chicago via Memphis. After your stay in Chicago, take the Texas Eagle, Lincoln Service, or Ann Rutledge to St. Louis, followed by the Ann Rutlege or Kansas City Mule to Kansas City. From there, I don't know where you want to go in Kansas, but from Garden City, Dodge City, Hutchinson, Newton, Topeka, or Lawrence, and from Kansas City, MO (Across the river from Kansas City, Kansas), you can catch the Southwest Chief, which you would take to Kingman, where (if it still exists) you can catch a thruway bus to Las Vegas, From Vegas, you can take a bus to LA, afetrwhich you take the Coast Starlight to San Francisco. Finally, you take the California Zephyr to Salt Lake City, and then on to Chicago, where you can take a train to your ultimate destination.
Hope this helps!
I'd go somewhat differently, but it's the same means to the end. Before I get into it, however, I don't know how familiar you are with Amtrak and its connection policies. A number of spots in the below suggested itinerary have connections between trains and Amtrak-chartered connection busses (sorry, some of it, like Vegas, is unavoidable, as there's no train service there). In each of the connections I list below, Amtrak guarantees that you'll make the connection. That means that if a train or bus is late and misses the connection, Amtrak will either put you up for a night in a hotel to catch the next day's service, rebook you on a later service, or provide alternative transportation (which is somewhat more rare).
So, my recommendation....
1). Start in Savannah and take one of the three Silver Services (Palmetto, Silver Star (the only one with a same-day connection), or Silver Meteor) to DC. If you plan to stay over in DC and are interested in seeing the southeast by train during daylight hours, the Palmetto is probably your best option, as it's a complete day train. I highly recommend reserving business class accommodations (as opposed to coach) for the long trip, however. They're comparable to first class seats on an airline with faux leather, free sodas and water, and a car attendant. Well worth it.
2). From DC, take either the Capitol Limited (fastest, but not entirely scenic) or the Cardinal (slow, but very, very scenic) to Chicago. Note that the Cardinal departs Washington only on Sundays, Wednesdays, and Fridays. I notice Washington isn't on your list, so if you plan on not stopping over in DC and would like to catch the Cardinal directly from the Florida train, note that you'll need to detrain from the Silver Star in Richmond, Virginia, catch a short 45 min bus connection to Charlottesville, and board the Cardinal there at 1:55 PM (and you'll still see the scenery, which is west of Charlottesville on the way to Chicago). If you plan to take the Capitol Limited to Chicago, just take the Silver Star all of the way to DC and change trains there.
3). From Chicago, head to Salt Lake City on the California Zephyr (this ensures prime hours for viewing the spectacular Glenwood Canyon and high desert in Colorado; going east (as referenced in AmtrakCrescent's itinerary) is risky as to whether the train will be on-time enough to see it all).
4). Visit Salt Lake City, then take the Zephyr (again) heading westbound to Emeryville, connecting to a bus to San Francisco.
5). From San Fran, head south to LA on the Coast Starlight (again, wonderful scenery if it's on-time).
6). Now, getting from LA to Vegas has a multitude of options, but unfortunately, none of them are rail. Amtrak has a number of bus connections straight from Los Angeles to Vegas, however, and I'd just take one of them (it's a little over a 5 hour ride on a tour bus, which I think is very do-able). Other options include renting a one-way car, catching a short plane trip, or going with another charter bus service other than Amtrak.
7). From Vegas, there is a 10:00 PM thruway bus connection from Vegas to Kingman, AZ (arrives 1:30 AM), which connects directly to the eastbound Southwest Chief at 2:03 AM. The Chief, while not quite as scenic as the Zephyr, Starlight, and Cardinal, has its own charm and is one of Amtrak's best trains in terms of on-time-performance.
8). When you say you want to visit kansas, I assume you mean Kansas City, which is a major stop for the Chief, arriving at 7:26 AM. Other Kansas stops include Garden City, Dodge City, Hutchinson, Topeka, Newton, and Lawrence. Regardless, you're going to want to end up in Kansas City at the end of your Kansas stop, which is where you'll want to board one of two Kansas City-St. Louis trains: the Ann Rutledge or the St. Louis Mule. The Rutledge leaves Kansas City at 12:30 PM and arrives in St. Louis at 6:25 PM (which is the train I'd suggest since delays are abundant on this route and you will want to arrive in St. Louis before bedtime, I'd imagine). The Mule leaves Kansas City at 4:30 PM and arrives in St. Louis at 10:25 PM.
9). From St. Louis, you're going to want to catch a connecting bus to Carbondale in order to catch the southbound City of New Orleans. The bus leaves St. Louis at 11:00 PM and arrives in Carbondale, IL at 1:00 AM. The City of New Orleans leaves Carbondale at 1:26 AM and arrives in Memphis at 6:27 AM.
10). You'll leave on the same train after visiting Memphis, arriving in New Orleans the same day at 3:32 PM.
Now, that pretty much covers all of the cities you list in one nice, big, circular route (sort of). Roswell's not on there, however, and that's because Amtrak doesn't serve that city with any sort of service, be it bus or train. Frankly (and I don't mean this in a bad way to any Roswell folks here), my understanding is that there isn't all that much to see in Roswell, at least not for making a major diversion. If you were to go to Roswell, my advice would be to catch a bus or drive from Vegas (3 hour trip), but that means you either need to drive back up north to catch the Chief, or drive south to catch the Texas Eagle/Sunset Limited out of El Paso Texas, which runs up to St. Louis. If you'd like more details on that version of the itinerary, let me know.
Hope that helps...
-Rafi
