Charlotte, NC
South Station is the Amtrak station in Boston that is closest to Copley Square. The T, the Boston subway system, connects South Station with Copley Square, but one change of train within the subway system is required. Hmm, I don't think so. Back Bay station is practically across the street on Google maps.
There are multiple train stations located in South Carolina. The cities of Charleston and Florence both boast Amtrak train stations to catch trains from.
Jesup, GA. From there you can take the Silver Meteor north to Savannah, Charleston, Richmond and New York or south to Florida.
The closest Amtrak station to Blue Ridge, GA is in Gainesville, GA. From there, you can take the Crescent north to Washington, DC and New York or south to Atlanta or New Orleans.
Albany, OR. You can take the Coast Starlight north to Portland or Seattle or south to Los Angeles. You can also take the Amtrak Cascades service between Seattle and Eugene.
South Station
Unfortunately, there is no Amtrak train service to Nashville. The old Union Station is now a Wyndham hotel and that is available for viewing. The closest Amtrak station is in Newburn, TN. From there you can take the City Of New Orleans north to Carbondale and Chicago or south to Memphis or New Orleans.
The closest Amtrak Station to Montgomery, AL would be Birmingham, AL which is served by the Crescent. It is 1 hour and 28 minutes away or 90 miles.
Tampa. On the Silver Star, you can go south to Miami or north to Orlando, Savannah, Raleigh, Richmond or New York.
VIA Rail Canada, in cooperation with Amtrak, operates the daily Maple Leaf from Toronto. The first Amtrak, and therefore US, train station is in Niagara Falls, NY, The train continues east and south to Buffalo, Rochester, Syracuse, Albany and New York.
Florence, SC is Amtrak's Myrtle Beach stop. From there you can take the Palmetto north to Richmond or New York or south to Charleston or Savannah. You can also take the Silver Meteor north to Richmond or New York or south to Charleston, Savannah. or Florida.
The closest Amtrak Station to Sioux Falls would be St. Cloud, MN, at a distance of 222 miles. The next closest would be either Fargo, North Dakota at 245 miles or Staples, Minnesota at 250 miles. Midway Station in the Twin Cities (Minneapolis/St. Paul, Minnesota) is 273 miles from Sioux Falls. Based on travel time, Fargo would be the fastest as travel is along I-29 all the way between the two cities.