Domestic slaves were used for cooking, cleaning, and caring for children. They were usually overseen by the woman of the house. Some slaves had very specific duties; for example, the ornatrix was the slave who was responsible for styling his mistress's hair.

Cooking, cleaning, body guards, chariot racer, a bath assistant at the bath houses, messangers....well, basically anything. One thing that they did do often, though, was carry water from the well to their master's house. they liked to gather around the wells and gossip about their masters or other slaves or the gladitorial games or whatever. That is, if their masters weren't wealthy enough to have their own supply from the aqueducts.

Erika's answer was a good one.
Here is another perspective of slave life in Rome.
This is the perspective if the slave were a Christian. How were they to accept their role in being used as a slave?
In the Roman Empire slaves were very numerous, with individuals owning hundreds and even thousands of slaves. The institution of slavery had the protection of the imperial government. First century Christians did not take a stand against governmental authority in this matter and advocate a slaves’ revolt. They respected the legal right of others, including fellow Christians, to own slaves. That is why the apostle Paul sent back the runaway slave Onesimus. Because he had become a Christian, Onesimus willingly returned to his master, subjecting himself as a slave to a fellow Christian. (Phm 10 to 17) The apostle Paul also admonished Christian slaves not to take improper advantage of their relationship with believing masters. He said: “Let those having believing owners not look down on them, because they are brothers. On the contrary, let them the more readily be slaves, because those receiving the benefit of their good service are believers and beloved.” (1Ti 6:2) For a slave to have a Christian master was a blessing, as his owner was under obligation to deal righteously and fairly with him. see Eph 6:9; Col 4:1.
The acceptance of Christianity by those in servitude placed upon them the responsibility of being better slaves, “not talking back, not committing theft, but exhibiting good fidelity.” (Tit 2:9, 10) Even if their masters treated them unjustly, they were not to render inferior service. By suffering for righteousness’ sake, they imitated the example of Jesus Christ. (1Pe 2:18 to 25) “You slaves,” wrote the apostle Paul, “be obedient in everything to those who are your masters in a fleshly sense, not with acts of eye service, as men pleasers, but with sincerity of heart, with fear of Jehovah. Whatever you are doing, work at it whole-souled as to Jehovah, and not to men.” (Col 3:22, 23; Eph 6:5-8) Such fine conduct toward their masters prevented bringing reproach upon the name of God, as no one could blame Christianity for producing lazy, good-for-nothing slaves. 1Timothy 6:1.

Of course, a slave’s ‘obedience in everything’ could not include disobeying God’s law, as that would have meant fearing men rather than God. Wrongdoing by slaves, even when committed at the direction of a superior, would not have ‘adorned the teaching of their Savior, God,’ but would have misrepresented and disgraced this teaching. (Tit 2:10) Thus, their Christian conscience would govern.
In the Christian congregation all persons, regardless of their social status, enjoyed the same standing. All were anointed by the same spirit and thus shared in the same hope as members of one body. (1Co 12:12, 13; Ga 3:28; Col 3:11)
While more limited in what he could do in spreading the good news, the Christian slave was not to worry about this. If granted the opportunity to gain freedom, however, he would take advantage of it and thereby enlarge his sphere of Christian activity.
1Co 7:21- 23.