Did every culture have a different date system? Did ordinary people bother to know the year they were born? How did traders from different cultures express the year and was there some commonlynused universal start date for these purposes even if culturally they recognized another start date (like how the Chinese have a calandar and the Jews have a calandar, both counting from different start dates)?
You may wish to look under the "calendar" section of the FAQ. These are two of the more useful threads for your question: 1, 2.
The short answer is that there wasn't one standard dating system and it was usually done generally on the basis of years of certain rulers. As an example, in the Middle Ages, lots of work was done to try and harmonize these different dating systems, so in universal chronicles multiple different systems are frequently set alongside one another. So, to give an example, when Otto of Freising dates the birth of Christ he notes:
This was in the fourty-second year of the reign of Caesar Augustus and the seven hundred and fifty-second year from the founding of the City [ad urbe condita], in the one hundred and ninety-third Olympiad, when five thousand five hundred years had elapsed since Adam, and an alien, Herod the son of Antipater, was ruling in Judaea, in the sixty-sixth week according to Daniel [...] Jesus Christ, the son of God [...] was born of the Virgin Mary.
Side note: there was no 0 AD. 1 BC -> 1 AD