Some good books to check out on this would be Greek Religion and Roman Religion both by Valerie M. Warrior.

Essentially, the Romans didn't adopt Greek religion. Some ideas where definitely passed and shared between the two groups. But on the whole, both groups' religions come from a common stock, thought to be from the Proto-Indo-European people. This can be seen in the chief deity of both - Jupiter and Zeus both have their etymological origins in *dyeu-peter- "God-father" or "Sky-father"