How did religion impact the Roman Republic?

Roman religious beliefs date back to the founding of Rome, around 800 BC. However, the Roman religion commonly associated with the republic and early empire did not begin until around 500 BC, when Romans came in contact with Greek culture, and adopted many of the Greek religious beliefs. Private and personal worship was an important aspect of religious practices. In a sense, each household was a temple to the gods. Each household had an altar (lararium), at which the family members would offer prayers, perform rites, and interact with the household gods. Many of the gods that Romans worshiped came from the Proto-Indo-European pantheon, others were based on Greek gods. The two most famous deities were Jupiter (the king God) and Mars (the god of war). With its cultural influence spreading over most of the Mediterranean, Romans began accepting foreign gods into their own culture, as well as other philosophical traditions such as Cynicism and Stoicism.
en.wikipedia.org/wiki/Roman_Republic#Religion

Religion in those days had no resemblance to what we call religion now. For one thing, going to meetings every seventh day was a Jewish thing. For another, the skies looked very different. The ancients (prior to the founding of Rome) were quite specific: they did not worship gods named for planets, they worshipped the planets. Jupiter and Saturn dominated the sky. By the time Rome came to be, worship of the planets was an inherited tradition, but still largely a family tradition.

Here is a careful investigation of ancient myths and legends, considering stories in hundreds of languages from all over the world and going back to 10,500 BC. It is very long, and it is still in progress: saturniancosmology.org/