Children 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 (the male head of the family), and never allowed back.
Both boys and girls wore a special locket, given to them at birth, called a bulla. A bulla was an amulet, a protective charm against evil. Girls wore their bulla until their wedding night, when it was set aside with other childhood things, like her toys. Boys wore their bulla until they were 16 or 17 and became a full Roman citizen, with the right to vote and hold office and marry.
Both boys and girls wore tunics. Boys wore tunics down to their knees, with a crimson border. In the home, girls wore a simple tunic with a belt at the waist. When girls went outside, the wore a tunic that reached their feet. Children were not allowed to use the public baths. They bathed at home or in the river.
Both boys and girls played with toys. Boys played war games, and had wooden swords, little soldiers, and chariots with wheels. Girls played with dolls, and dollhouses, and tiny sets of dishes. Both boys and girls played board and ball games, like tic-tac-toe and knuckleball (jacks).
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 and wives. A wealthy family could also adopt a plebian child. This happened when the patrician family had no children or heirs.
Most boys and girls were educated at home. If the family could afford it, boys might also attend school and study reading, writing, math, oration, and how to be a good Roman citizen. Educated slaves were often their teachers. Greek slaves especially were in high demand as teachers for Roman children. It was the woman's job to teach the girls how to be good wives and mothers. The paterfamilias (the oldest male in the family) was responsible for teaching all the younger males in the family both academics and trades, and also how to act in society.
Children had no rights, but there was a protective custom or system in ancient Rome. The paterfamilias (the oldest male in the family) 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.