Depends which year, which emperor and which Rome you are talking about mainly.
For The Great Roman Empire, most young men would join the army or work for their fathers, on other times when Rome became the pinnacle of the world men would become scholars or traders or even masons of the arts this is the same for most of the Roman Empires. While for a young lady, she would tend to chores around homes, raise children or on a degenerate note be a harlot. This is just depending on the family they were raised in or the class they were born in.