From my understanding, religion in the ancient world (or for a better word, 'Pagan' religions) had little to do with how you should/shouldn't behave, and didn't have a moral code to follow as such, and also seemed 'selfish' in a way that, it was about what YOU could receive. Animal and human sacrifices being a good example of this. Their idea of religion was (obviously) very different to Abrahamic religions and trying to grasp how people of the past saw the 'Gods', or how they defined religion, is very difficult from a 21st century point of view.
Is there anything available to us from the past which could help your typical modern person understand how religion was viewed in times gone by?
A good hint is embedded in the ancient languages themselves -- ancient Greek has a set of words specifically dedicated to the system of χάρις, kharis, which loosely translates to "that which is delightful".
The ancient Greek religion revolved around a system of reciprocity in worship: humans would offer sacrifices (kharienta or kekharismena - sorry for only offering transliterated versions, the original Greek letters aren't in my notes!) to the gods, and the gods would then in turn dispense khariessa (blessings, gifts, etcetera). This system was the lynchpin of their religion.
Your understanding is fundamentally correct - someone in the ancient Greek (or Roman) faith could commit some huge kind of misdeed, but immediately get back into the good graces of the gods through appropriate kharienta.
Essentially, the point I'm trying to illustrate is that this system was an ingrained part of everyday life; the concepts of sacrifice and blessings were so inseparable and intertwined that the Greek terms for them are just so damn similar to each other.
SOURCE: wrote an undergraduate paper which argued that the kharis system (among other things) rendered Aristotle's theos inappropriate for worship in Ancient Greece. The text I referenced for this section in the paper is
Gill, C., Postlethwaite, N., & Seaford, R. (1998). Reciprocity in ancient Greece. Oxford: Oxford University Press.
Edit: "each other" Edit 2: fixed ambiguity in source.