Rome wasn't Christian until 326. And the Gregorian calendar wasn't established until the 500s. So what year did people think it was when they switched? Did anyone ever know it was year 1?
In short, many different dating systems were used before the Anno Domini system was adopted. Common systems included a king's regnal years, the consuls of Rome (until the office fell into disuse), or the "Year of the Martyrs" dating from the ascension of Diocletian, a significant persecutor of Christians. This chaos probably helped Dionysius's Anno Domini system be adopted.
You might also be interested in these previous threads about the calendar, from the subreddit wiki.
There's no one date. Some credit Bede with popularizing the AD system in the West, but Dionysius Exiguus proposed it circa 525.
If by dating system you mean calendar, again each region was different but the Gregorian calendar was introduced in 1582.