The short answer for this is that Woodrow Wilson was the first sitting president to travel to Europe when he joined the Paris Peace Conference and the Treaty of Versailles. The intent of his trip was to push his The Fourteen Points at the conference in order to create a sustainable peace with a democratic institution--the League of Nations--mediating global events.

Wilson travelled to France on the George Washington with his wife in December 1919. They were welcomed with large crowds carrying signs reading "Hail the champion of the rights of man" and "Wilson the Just." Wilson travelled around England and Italy before the conference but the secondary sources I'm reading from don't mention if he met with heads of state before the conference. It does say that reporters travelling with the President wrote home about the large crowds greeting him at every stop he made.

Needless to say, Wilson did not fly to the conference. While Roosevelt was the first President to fly in 1910, Wilson's trip to Europe in December came some six months after the first transatlantic flight by British aviators John Alcock and Arthur Brown in June of the same year.

So to answer your question: Wilson's trip is probably the first "large scale foreign trip" and these would become relatively common place by the time FDR was meeting up with Churchill and Stalin--again by boat--so these trips were more or less common before flight was a safe form of transportation.

My knowledge of this comes from Standing at Armageddon by Nell Irvin Painter.

The first US President to travel abroad while in office was Theodore Roosevelt, who visited Panama in November 1906 to inspect the Panama Canal construction. So for the first 117 years of America's history, no sitting president ever left the country. Roosevelt's successor, William Howard Taft, also visited the Panama Canal Zone as president-elect, and Mexico as President. After that was Wilson, who made two lengthy trips to Europe, about two months long each. Since then, every president has traveled abroad while in office, some more than others, with the exception of Herbert Hoover, who did travel as president-elect but not while president.

Source: US State Department, Office of the Historian.

The first president to travel extensively by air (in fact, the first president to fly while in office at all) was Franklin Roosevelt, who visited Europe by Boeing 314 flying boat and used a C-54 Skymaster for domestic US travel. Since his presidency, every president has traveled by air. President Eisenhower was the first to use the classic Boeing 707-based Air Force One aircraft in 1959.

Did Presidents send (for example) telegrams to other world leaders in the 19th century? Was there anything similar to a classified telegram, and how did that work? Was there a White House telegram station or would the President and his staff have used public facilities?