Opening Day: Oct. 27, 1904
In the fall of 1904, the new subway system was completed. Anticipation was high.
"Man in whose keeping millions of New-Yorkers will begin this week to put their lives"
Frank Hedley, General Manager of the IRT, on the front page of an illustrated supplement to the New-York Tribune, Oct. 22, 1904, 5 days in advance of the subway's opening (fig. 8). 
It's probably merciful that the New York Tribune offered only "Kernels of the speeches" (fig. 9) offered on the first day of service, as apparently the long speeches contributed to a 30-minute delay in the inaugural ride getting underway, with Mayor George B. McClellan, Jr. serving as motorman. So, service delays have been with the system from the very beginning.
Note that the schedule indicates 24-hour service, from the beginning. 
The New-York Tribune went on to describe the scene in sensational, perhaps even purple prose (fig. 10):
"Birth of the Subway Crush"
"Indescribable scenes of crowding and confusion, never before paralleled in this city..."
"The old six o' clock Brooklyn Bridge car crush paled into insignificance when when contrasted with the deadly, suffocating, rib smashing subway rush..."
"... the presence of the police alone averted what would undoubtedly have been panic after panic, with wholesale loss of life."
"Two thousand... residents, who wished to go downtown, and those who had ridden uptown and wished to return, then surged and swarmed madly at the downtown entrance, frantic to descend." 
The New York Times' coverage included this charming description of the soundscape, as the inaugural train passed over the Manhattan Valley Viaduct at the present-day 125th St. station:
"This viaduct was the only section where the public could view the first train, and a great crowd had assembled at every available place of vantage for blocks around. On the vacant lots, on roofs and in every window, men, women, and children cheered and waved their hats or handkerchiefs. Mr. Morrison [the official motorman] slowed down and started the whistle. Every factory in that part of the city answered him, and the boats out on the river began a tooting that could be heard in the train until it dived into the earth again at One Hundred and Thirty-third Street." 
 New-York tribune. (New York [N.Y.]), 23 Oct. 1904. Chronicling America: Historic American Newspapers. Lib. of Congress. <http://chroniclingamerica.loc.gov/lccn/sn83030214/1904-10-23/ed-1/seq-29/> Accessed on July 22, 2015.
 New-York tribune. (New York [N.Y.]), 28 Oct. 1904. Chronicling America: Historic American Newspapers. Lib. of Congress. <http://chroniclingamerica.loc.gov/lccn/sn83030214/1904-10-28/ed-1/seq-1/>. Accessed on July 22, 2015.
 New York Times (1857-1922); Oct 28, 1904; ProQuest Historical Newspapers: The New York Times with Index, pg. 5. Accessed on July 22, 2015.