Though the first Transcontinental Railroad was finished in 1869, the Overland Route seems to have only taken passengers between San Francisco and Council Bluffs, Iowa / Omaha, Nebraska via the Overland Flyer. Was there ever a train that ran straight from San Francisco to New York City, or for that matter, from any city that could be considered decidedly on the Pacific to one that sat on the Atlantic?
I guess the first one could be the Pullman's promotional train of 1870. It started from Boston on May 23rd, 1870, went to San Francisco and then back to Boston, completing the round-trip on July 1, 1870. It was not a regular line. Specifically built for the trip, it took 130 passengers, mostly wealthy members of the Boston Board of Trade with families.
http://cprr.org/Museum/Trans-Continental/index.html - the newspaper published aboard this train
Edit: Obviously, it had stops on the way!
Edit2: I understand the question is about the single train, and not concerned if the train continuously went all the way on the tracks (without a ferry).
From 1993 to 2005, Amtrak's Sunset Limited operated between Los Angeles and Florida, initially to Miami and later cut back to Orlando. Hurricane Katrina in 2005 damaged the CSX railroad line between New Orleans and Jacksonville, and the train was suspended east of New Orleans. The chronology of service appears on the Wikipedia page.