First - the Mediterranean religions were very different from each other, although all were polytheistic. Some had dedicated priests, some did not (Rome had both dedicated priests and a role for male heads of families, who led worship of the household gods). Some had a focus on individual devotion (most notably in Syria), some were more communal. Socially, worship was an occasion for an expression of civic virtues and often a celebration of significant events (foundings, victories), plus benefaction and display of status.

The gods ranged from local powers of a spring or grove through (in Greece) personifications of emotion or capability (Love, Memory) to supernatural aspects of society (the Avengers) or cults devoted to particular activities (War, Agriculture) to sources of law and justice (Zeus the Law-Giver). How you conceived of them and what you did were highly contextual. Priests knew the right way to conduct the rituals and the appropriate prayers, so were experts on the care and feeding of the gods. They often had authority at certain times and occasions, and sometimes their persons were sacred (as in Vestal Virgins). The rites could be arcane - one High Priest in Rome had to sleep on a bed in contact with the earth and could not see a man in chains (if he did the man was immediately freed).

In short - the world was numinous, everyone acknowledged this and did their best to placate the invisible powers, and some people were professionals at this.