The Boeing 707 entered service in 1957, and is probably most associated with the start of the jet age.
Most commercial flights were likely propeller-driven aircraft through the 1950s, but some jet travel had begun by the end of the decade.
Edit: I've spent some time trying to find stats on flights / passenger miles by aircraft type for the 1950s & 1960s without luck. That'd be an interesting add to this thread if anyone has the information.
I do find total passenger and freight data here: <https://transportgeography.org/contents/chapter5/air-transpo...>
As of 1960, both categories were effectively nil. Passenger travel hit 1,000 billion passenger-km around 1980, the pre-2020 peak was about 8,500 billion passenger-km. Data based on IATA.