Just among currently operating airlines… KLM 1919, Qantas 1920 Aeroflot (and several other European flag carriers) 1923, Delta 1925
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.