I was recently reading Torah (Shemos 12:2), in which Hashem established the first month with Moshe. How does this coincide with the Roman calendar? How do we know that we are observing Shabbat on the proper day?

The Roman calendar (both the Julian calendar and the Greggorian calendar) is a solar calendar and the lunar months play no role. The 15th of a Christian month does not signify a full moon for example. The Jewish calendar is made up of a year consisting of 12 lunar months with seven leap years of 13 months within every 19 year period. We know which day is shabbat based on historically confirmed, unbroken tradition.