August 12 Monday – The men left Baden Baden by rail and arrived at Lucerne, Switzerland where they joined Livy, the children and the rest of the party who had been there a few days.

The Clemens party left Heidelberg on 23 July 1878 and proceeded to Baden-Baden, where they would meet Joe Twichell on 1 August.

Twain hired the portier of the Schloss Hotel, Georg Burk, as courier for $55/month and expenses and departed Heildelberg by train for Baden.

There are two possible railway lines that could have been used for this trip:

Sam and his entourage slowly migrated to the southwest over the next week, overnighting in Hanover, where Sam and Clara Spaulding attended a performance of the comic opera Roberto il Diarolo; Gottingen, Kassel, and Wilhelmshéhe; and Frankfurt, where the family toured the Goethe Geburtshaus in the Altstadt and again crossed paths with Frank Mason, the U.S. consul general there and former editor of the Cleveland Leader.  [ Page 249 The Life of Mark Twain - The Middle Years 1871-1891]

It's likely Twain traveled to Indianapolis from Paris, IL on January 1, 1872 as he attended church in Paris December 31, 1871. 

Both the New York Central and the Pennsylvania Railroads had connections between Paris and Indianapolis.  The NYC, as per the Wikipedia article, maps the Terre Haute Railroad and the St Louis and Indianapolis Railroads.  The Wikipedia article for the Pennsylvania Railroad maps the Paris and Terre Haute Railroad and the Terre Haute and Richmond Railroad. This line was associated with the Vandalia Railroad.

Twain traveled from Newark, NJ to Oswego, NY sometime between November 29 and December 1.  A likely itinerary would have been to take the Central Railroad of New Jersey from the Communipah Terminal, through Elizabeth to Hampton.  From Hampton to the Delaware River, the Warren Railroad and then the Delaware Lackawanna and Western through Scranton to Hallstadt and then to Binghamton.  The  DL&W no longer required the New York and Erie to reach Binghamton; the Syracuse Binghamton and New York Railroad to Syracuse;  and the Oswego and Syracuse Railroad to Osweg

The November portion of Mark Twain's Three Speeches Tour begins in Boston, Massachusetts, 

November 1 - Music Hall, Boston, Massachusetts - "Artemus Ward"

Mark Twain departed New York City October 14, 1871 traveling to Bethlehem, PA.  The trip would be aboard the New Jersey Central  and the Lehigh and Susquehanna Railroad, which was leased to the CNJ  March 31, 1871 (New Jersey Central).