Hi,
Your best bet would be to visit amtrak.com and select the map/route page. This shows the skeleton of the Amtrak service.
Often surprising for tourists is the lack of Amtrak service to many places, some states have no service at all.
Having said that, I have enjoyed many tens of thousands of miles of Amtrak train travel on my visits to America...
All of your cities are reachable by train, although, as mentioned, the last furlong might entail an Amtrak motor coach transfer.
America is a huge country, and the routes are often only one train a day.
Do be sure to check the timetables carefully, as a trip that looks at first glance like a 12 hour ride might be a 36 hour one!
If you have a look at the Amtrak .com pages, you can plan which routes work best for you.
We are all Amtrak geeks on this site, so if you can list your start and finish cities, you can be sure we will suggest routes for you.
As you have the cities listed above, if they are your desired route, they are more or less in a circle and that sequence should work. As others have mentioned, there is no longer a connection from Florida to New Orleans, although there was some years back. If I were to start in Florida, I would take the train up to Washington, and then from there down to New Orleans, and so on... but I am a train fan, so maybe flying from Florida to New Orleans would be the easy start option, then all rail from there...?
Please feel free to ask for more specific advice, and also do look into the "rail pass" options which might work out cheaper.
The main choices on the trains themselves is whether to travel in a private "bedroom" or to use the regular seats, known in America as "coach" .
Although all advice given here on the site is well meaning, do your own research too, as even we geeks often can't agree amongst ourselves!
Bon Voyage!
Ed.