Generally Roman and Greek people in classical antiquity would worship the God(s) who they wanted favours from or who might grant them protection. So if you were going to War you might burn an effigy and make a sacrífice to a God that could lend a hand (Zeus, Apollo, Ares), if sailing at sea (Neptune).

Some Gods like Bacchus were physically harmful, and people would hang phallic totem outside their doors to ward off Bacchus from breaking into their home and sodomising them (literally).

The Gods were generally troubled and fallible like people, but had special powers. Moral virtue was something that living people taught, the philosophers for instance.

The Romans had soothsayers who could predict the future based on the reading of natural phenomenon (flight pattern of birds, congealed blood of slaughtered beasts). The Greeks had oracles (mad woman) who talked in riddles that required imaginative interpretation. They had the vestal virgins protect the holy eternal flame (Rome). But as far as I know they did not have priestly class who had a working relationship with the divine, if that's what you are asking.