Laying the Groundwork
Construction for the IRT subway officially commenced on March 24, 1900, in a ceremony in front of City Hall, featuring then-Mayor Van Wyck breaking ground with a ceremonial shovel. 
Discussion and plans for various ambitious rapid-transit rail systems had been underway for decades. Other cities had already constructed subway systems, notably London (1863) and Boston (the first subway in the United States, 1897).
Manhattan's first attempt at an underground subway, by Alfred Ely Beach in 1870, was a peculiar and secretly-built project, constructing a short stretch of track in lower Manhattan (312 feet, under Broadway between Warren and Park Place), operated by pneumatic power. According to Brian Cudahy, it was undertaken outside the control of "Boss" Tweed and the Tammany Hall political machine, and thus was doomed to be cut short, despite its inventor's plans for expansion. We can wistfully yearn for this early incarnation of the subway, as it was to feature a live piano player on the platform, to prevent the gloomy associations of being underground, as well as including frescoes, fine upholstery, and a fountain. 
Clifton Hood identifies two distinct periods in the politics of the New York City subway system; the first from 1888-1907, where business interests were the dominant force in decision-making, and the period after 1907, when "professional politicians made the most important transit decisions"  A detailed accounting of the history leading up to the inception of the IRT subway project is beyond the scope of this focused look at Morningside Heights, but excellent histories are provided by Cudahy (1995) and Hood (1995) (see notes to these pages for full citations), and a more historically-contemporary account is found in the introduction to the Interborough Rapid Transit Co.'s own The New York Subway: Its Construction and Equipment from 1904, the year of the first IRT lines' opening. August Belmont, a wealthy investor, was a key figure in consolidating the Interborough Rapid Transit Co., and providing the much needed capital and financial structuring to allow the work to proceed.  Fig. 3 shows an illustration from the introduction to the IRT's New York Subway report mentioned above, which gives a feeling for the sense of pride in this grand public works project, and a vision of its integration as a key thread in the fabric of the city.
The map above (fig. 4), from the 1904 "New York Subway Souvenir", shows the final route of the first IRT line, along with a profile of the terrain through which the construction passed, heading north from City Hall to Grand Central Terminal, and then making a cross-town turn to the west side of Manhattan, before continuing uptown.
Two methods of construction used in the IRT subway construction were deep-bore tunneling, and shallow excavation, the latter often referred in this context as "cut-and-cover". Deep-bore tunneling had been used widely in the London Underground's construction, but IRT engineer William Barclay Parsons (Columbia College, 1879; Columbia University School of Mines, 1882) considered that the "cut-and-cover" method might prove cost-efficient in the long run, since it would avoid the need for a system of elevators to access the stations.  The obvious disadvantage was that it caused considerable disruption on the street in busy areas. To this New Yorker, it seems amazing that the work could have been achieved without riots.
Fig. 5 shows a photograph illustrating the "cut-and-cover" technique, with shallow excavation from street level to a track bed which would then be re-covered. This view is at 117th St., looking north. Note the arch in the tunnel. This is a transition to another construction method, a concrete-lined arched tunnel, which was used to proceed north to the 125th Street station, where yet another engineering method had to be called upon to accommodate the changing terrain.
Although there was less street and residential disruption to worry about, compared to lower Manhattan, the terrain in upper Manhattan posed challenges. The Morningside Heights stretch of the IRT provides a good illustration of this. We've seen an illustration of the "cut-and-cover" technique, and the concrete-lined arched tunnel, in fig. 5; heading further north, the terrain changed yet again, and forced construction up out of the ground. In fig. 6 we see the Manhattan Valley Viaduct under construction. This elevated viaduct, when completed, included the 125th St/Broadway station of the first-phase IRT line.
This view is looking to the west, along what I'm pretty sure is 125th St. (due to the streetcar tracks visible along the road), but I've seen some photos of this view identified as 129th St.
Fig. 7 shows a photo of the completed Manhattan Valley Viaduct, from a similar vantage point as fig. 6.
 Joseph Cunningham and Leonard De Hart, A History of the New York City Subway System, New York?: J. Schmidt, R. Giglio, and K. Lang, 1993, 19.
 Cudahy, xiii.
 Cudahy, xvi.
 Clifton Hood, 722 Miles: The Building of the Subways and How They Transformed New York, Baltimore: Johns Hopkins University Press, 1995, 13.
 Interborough Rapid Transit Co. The New York Subway: Its Equipment and Construction, 1904, https://archive.org/details/interboroughrapi00interich, accessed on July 21, 2015.
 Cudahy, 23.