There was not a tremendous change in family life and slavery from the Empire and the Republic the primary change was political, and social. The growing affluence of the Empire led its people to call for more and better sources of entertainment and diversion. This need was met by gladiatorial fights and the Circus Maximus. Additional diversions were theaters and the Roman baths.
Roman society during the Empire was strongly patriarchal. The oldest family male, either the father or grandfather, was the paterfamilias. He could divorce his wife at will, legally kill her if she committed adultery, could also kill his children at will or sell them into slavery if he wished. He could force them to marry when and whom he wished, even against their will. Still, important decisions were made by a family council consisting of all adult males. Roman wives were expected to exemplify the virtues of chastity, modesty, fidelity and dedication to the family. It was she who supervised household slaves.
Slavery actually expanded under the Empire. At one point, one third the population of the Empire were slaves. Slavery was not based on race or color; in fact black or Germanic slaves were considered quite desirable, and conspicuous indicators of wealth. Household slaves Worked as craftsmen, shopkeepers, servants, even business agents. Many lived quite comfortably; and could expect to be manumitted (freed) at age thirty if they had been faithful. Unfaithful servants could be punished at the master's whim, including execution for serious offenses.
Slaves in the countryside were not so fortunate. They worked on large latifundia,as state owned slaves. Others were required to work in quarries or road building. Quite often they were chained togeher in teams to work. The work and their living conditions was so harsh that several slave revolts resulted. The most famous revolt was led by Spartacus which required eight Roman Legions to defeat. At its end, Spartacus was crucified for his efforts.