Comparative religion degree here, I'll try to put that to some use.

Generally, you would learn these practices from family and what you know of the stories in Scripture would come from the public reading of the Torah Prophets and Writings in the synagogue, in the form of texts like the Targhums.

Targums were Aramaic translations ( lingua Franca at that time) that are combined together with traditional commentary. There would likewise have been Greek versions if you were Hellenistic jew.

It's also important to know that some common Bible stories had illustrations in Hellenistic synagogues like the one at Dura Europos, and you could learn the stories from there too.

As for how the laws would be observed, that would depend on your family and what sect you belonged to.

For example if your father was from the Priestly caste, you would likewise be a member and learn that role from your father.

If there was a law that you needed help with observing, you would either ask a priest, or if you are of the Pharisaic sect you would look at the traditions of the fathers as passed down by a teacher, iE rabbi.

A festival like say the Festival of Tabernacles, booths IE Sukkot is observed slightly differently by different sectarian communities.

The Samaritans for example build their Sukkah in a way different then rabbinic Jews do today. Different sects in second temple times like the Dead Sea sectarians likewise observed the same laws but to varying degrees of strictness, had different interpretations that led to slight differences in practices.

Professor Lawrence shiffman has some really good books on the Dead Sea Scrolls and the differences in in Halacha between the Dead Sea sect and Orthodox Jews today. Very interesting stuff.