Despite how powerful the personal expression and understanding of the gods may be, and it is indeed very much so, the family is the center of Greco-Roman religious life, and this means that there is a big emphasis on the familial religious duties and responsibilities. Religion is not a thing external to the family, but part of its core and roots.
The complex relationship between honoring the gods and the nature of the household, and the family unit itself, makes honoring the gods inseparable from honoring one’s home and places a huge importance on ancestor veneration. We honor those who came before us as a means of honoring the influence of the gods that have provided the continuation and blessings of our family. The gods who have provided offspring, (such as Leto who graces mothers with motherhood, Hera who brings for lawful heirs, and Aphrodite who is by her very nature of attracting unions is generative), and those who nurture the children born into the family and protect them (such as Apollon, Artemis and Hekate). These are the gods who have blessed the marital unions that have led to the expansion of our families (Zeus Teleios and Hera Teleia). We recognize too that our ancestors also had personal complex relationships with the god(s) that they worship which benefited the health and welfare of the family. They have offered praise and prayed over sick children and grandchildren. They have tended to the health and welfare of the family with the grace of the gods. So we honor our ancestors who embody the love of the gods in our lives and in our families. Those who protect our families, those who provide comfort in our families, those who fed the members in our families and so forth. We honor them that they continue to care for us, their descendants, as they did in life.
We also recognize that the household itself is alive with the blessings of the gods and welfare that they provide our families and have provided. The very house itself becomes a thing pertinent in our religious life. More so in cases of ancestral homes that have been handed down through the generations I think. Nowadays we have significantly less ancestral attachment to our homes as we do not feel the ties to the home for the birth place of generations. It may even be hard for some to imagine. But even still the gods are imbued in the physical structures of our house where they reside and bestow their blessings. Hestia at the hearth, Zeus at the center and in the courtyard (if your home has such a thing which is rare in this period), Apollon in the foundations and at the entrance with his twin, Hermes, Herakles and Hekate, Poseidon holding up the walls, the Dioskouri upon the roof. It is by tradition that the gods were honored at these places as honoring the gods who dwell within the house. The blessings are bestowed through these centers of the house. For instance the Herm and Apollon Agyieus stone was literally worshiped outside the entrance of the house. Hekate was honored before the doors within the home, protecting the sanctity of the house which served as the very heart of the family.
Ancestors and household aside, the familial religious life also is part of how we treat the members of our families, the support, love and devotion to our families. It is part of how we are expected to treat our parents, our grandparents, our brothers and sisters, our children and stepchildren (for the latter we can look at mythic example by telling us how not to treat our children and stepchildren through tragedies and other media. To betray or otherwise render some assault on our family is to dishonor the laws of the gods, and dishonors the gods themselves who care for and provide for our wellbeing and the wellbeing of our families. They who reared us within the families that we have been allotted. This matter was taken so seriously in ancient religious thought that a number of the Delphic Maxims deal with familial responsibilities and interrelations.
Therefore our religious duties extend to honoring our families, the living members and deceased members as part of honoring the gods. To honor them in a fashion (and to show what love, affection and respect we have for our families and ancestors) honors the gods, and by honoring the influence of the gods in the household we again honor the gods in their greater common manifestations. This recognizes the gods in our daily lives in a more intimate and personal level, and it helps us not only establish such kharis between our selves and our families with the gods, but also enriches and informs personal devotional relationships with the larger governing nature of the gods n the world and cosmos.