Short answers, pulled from the story.
The core principle was do ut des, a Latin phrase meaning "I give that you might give." Roman religion was contractual: correct prayers, rites, and sacrifices obligated the gods to reciprocate with protection and favor. It was based on knowledge and correct practice rather than faith or dogma.
The Vestals were a public priesthood of six women devoted to Vesta, goddess of the Roman state's sacred hearth. They were the only women in archaic Roman society not required to be under the legal guardianship of a man. A Vestal who lost her chastity while in office was buried alive. Theodosius I extinguished Vesta's sacred fire and disbanded the order.
Augury was central to Roman public life. Augurs read the divine will through observations made within a sacred space called a templum, and their findings could adjourn all official business for the day or overturn proposed acts. Cicero, himself an augur writing in the mid first century BC, called the augur the most powerful authority in the late Republic.
Christianity became the official state religion of the Roman Empire in 380 AD. The emperor Gratian had earlier refused the office of pontifex maximus, beginning the formal dissolution of the old priesthood, and Theodosius I extinguished Vesta's sacred fire and vacated her temple.
Mystery religions were cults that offered initiates salvation in the afterlife through secret rites of passage, which might use dance, music, intoxicants, and theatrical effects. Conservative Romans viewed their exclusive oaths and secrecy as characteristic of conspiracy or subversion. The Senate moved to suppress the Bacchanalia in 186 BC, and astrologers were officially expelled from Rome at various times, including in 139 BC and 33 BC.
Rome's general policy was to absorb the deities and cults of conquered peoples rather than eradicate them, believing that preserving tradition promoted social stability. Rome built temples to local deities that framed their theology within Roman religious hierarchy. Inscriptions throughout the empire record the side-by-side worship of local and Roman deities, including dedications made by Romans to local gods.