For Romans, family was the most important thing. The whole family would all live together in one house or apartment. The family included all unmarried sons and daughters, as well as married sons and their wives. Married daughters went to live with their husband's family.
The family was ruled by the paterfamilias. (Also spelled pater familias) This was always the oldest male in the family. Father, grandfather, uncle, oldest brother, whoever was the oldest male was the absolute ruler of the family. The paterfamilias owned all the family's property and had the power of life and death over every family member. The paterfamilias was also responsible for teaching all the younger males both academics and trades, but also how to act in society.
The paterfamilias was responsible for all the actions of the family. If someone in the family got in trouble, the paterfamilias had to pay the consequences. The paterfamilias could exile members of the family, beat them, sell them into slavery, even kill them with no threat of reprisal.
The paterfamilias was expected to treat his family with fairness and compassion and if he did not, that person would be shunned by the rest of Rome.
Under the kingdom, and then under the republic, women had no rights. A woman's role was to teach her daughters how to behave, and to bear and raise children. Under the empire women received some rights. They could own property, inherit, even get a paid job.
Children were loved. They were educated to the best of a family's ability to do so. They were allowed to play and visit friends. But they were also trained to obey elders. You never talked back to an elder Roman. You never talked back to your family. Doing those things could actually get you thrown out of the house, exiled by the paterfamilias, and never allowed back.
Romans did adopt children. If children were captured in a conquest, they were brought back to Rome. Some were made into slaves, but many others were adopted into Roman families and raised to be good Roman citizens. A wealthy family could also adopt a plebian child. This happened when the patrician family had no children or heirs.
In fact, you could be adopted into a Roman family even if you were an adult. Julius Caesar adopted Octavian, after he had proven himself in battle. He was to be Julius Caesar's heir. (Octavian changed his name to Augustus, and ultimately became the first Roman emperor, after Caesar was assassinated, and after a civil war placed him in power over the objections of several statesmen, including Cicero.)
Older people were treated with honor. The family respected the wisdom and experience that older people had. Within a family, the elders were allowed to work or play as they wanted. This is because Romans believed that the spirits of the elders would bother them if they were treated badly in life.
Most household slaves were well treated. Since they were property and cost money, they were given good care so that they could provide good work. However, they were property and could be sold. On the reverse side if they gave good service, they could be freed and even adopted into the family.