Alongside the state cult ran a domestic religion that most Romans met far more often. The Lares and Penates guarded the household and its stores, Vesta its hearth, and the genius of the head of the family received offerings on his birthday. Shrines have been found in houses across the social range at Pompeii and Herculaneum. Birth, marriage and burial were family business, the dead were tended at fixed points in the calendar, and burial societies and trade associations gave poorer Romans a funeral and a commemoration they could not otherwise have afforded.
Rome imported gods with little anxiety. The habit of identifying a foreign god with a Roman one (Jupiter with Zeus, Mercury with Gaulish and Germanic gods, Sulis with Minerva at Bath), meant that conquest usually produced fusion rather than replacement. The cults of Isis and of Mithras spread through the empire on that basis. Mithras is a standing warning about interpretation: the cult left hundreds of underground meeting rooms and a remarkably consistent iconography, but nothing written by its own members, so every account of what initiates were taught is a reconstruction from images.
Where Rome did act against a cult, the stated grounds were public order, secrecy, conspiracy, or practices held incompatible with Roman custom, rather than false belief. That is also the frame within which Christians were treated as a problem, since a refusal to sacrifice looked to Roman officials like withdrawal from a civic obligation rather than a theological position.
Almost none of this is known from Roman religious literature, because there was very little. The great antiquarian survey by Varro is lost except where later writers quote it, and much of what survives of it comes through Christian authors quoting it in order to attack it. The most direct evidence is calendars, priestly records, inscriptions and dedications.