The first commercial passenger service ran on 16th April, 1853 from Bombay.
But railways were used in the country long before that. I am not sure where they got the 1832 from, but as of now, the oldest documented railway known, is the RedHill Railroad in Madras, which was opened in 1836 by Sir Arthur Cotton, of the Madras Corps of Engineers. This line had extensive freight operations, occasional passenger runs, and locomotives and wagons that were designed and built in India as well.
In the same year (1836), he operated another...
more... short railway line near Trichy.
Between 1842-45, a short railway was operated in Visakhapatnam, then near Rajahmundry (both in Andhra Pradesh) between 1847-52 (Both railways were constructed by Sir Cotton). And a more famous railway, that was used in Roorkee from 1851.