Probably someone will find an earlier example, But the Soviet film Kosmicheskiy reys (Cosmic Journey) (1936) is mentioned on p. 21 of Reconsidering Sputnik as "the first Soviet film which showed weightlessness in action" (edit: after a little more checking, it looks like my earlier possible example below probably counts as a physically accurate depiction, and January First-of-May's answer is even earlier and sounds like it's accurate as well, but I'll leave this answer up in case people are interested in the earliest filmed examples), you can see it online here:
The launch begins around 27:40 in the video, with the astronauts earlier shown being put in liquid-filled tanks to help them withstand the initial acceleration. At 29 minutes in, the translation of the subtitles says "In the 5th minute of the flight Sedych switched off the rocket-fires and at that very moment the astronauts became weightless". So, seems to be an accurate understanding of the idea that you'd experience being heavier during the initial accelerating phase of a rocket launch, then as soon as the rockets stopped you'd be in free fall and therefore experience weightlessness.
An earlier depiction of travel to moon, Frau im Mond (Woman in the Moon) (1929) directed by Metropolis director Fritz Lang, did show weightlessness for part of the trip, but not when they initially turned the lever to stop the intense acceleration which was exerting huge pressure on them, weighing them down. It's possible the idea was just that they were continuing to fire the rockets at a lesser rate to cause 1G acceleration (edit: actually it seems pretty definite this was the intention, see my comments below), but p. 15 of The Spacesuit in Film suggests it might have been due to a misunderstanding of when weightlessness would occur:
Once the pressure is gone, crew members awaken, not yet weightless. (At the time, some believed that weightlessness would only occur when a ship passed from Earth's gravitational field into the Moon's field.) In later scenes they employ straps on the floor to walk about in weightlessness, and Gustave floats in the air. A frustrating attempt to drink a beverage is dealt with by scattering the liquid into floating bubbles which astronauts catch in their mouths.
The movie is available here:
The initial launch happens at 1 hour 34 minutes, then at 1:37:38 a character pulls a lever and the acceleration on a dial is shown dropping back down to about 10 m/sec^2 (about 1G). And note that a subsequent shot at 1:37:59 shows that there's still an exhaust trail behind the rocket, so I think it's actually more likely they understood that the only astronauts could experience 1G in flight was if the rocket was still firing, rather than making the mistake about the Earth's gravity suggested by the book above. Another reason I doubt they were making such a mistake is that the movie had a technical consultant, Hermann Oberth, who was an early pioneer of rocket science, and as described on p. 5 of this paper he had actually done experiments with weightlessness:
Oberth was born in the Transylvanian city of Sighisoara, Romania to a Saxon family. Around the age of 11 he became fascinated with spaceflight through the writings of Jules Verne’s, “From the Earth to the Moon” and “Around the Moon”, rereading the books to the point of memorization. He constructed a model rocket as a student when 14 and conceived a multistage rocket.
He was drafted during World War I into a German infantry battalion and in 1915 was reassigned to a medical unit in a hospital where he conducted experiments concerning weightlessness. He later resumed his rocket designs. In 1919 after the war he moved to Germany to study physics. In 1922 his doctoral dissertation, “By Rocket into Planetary Space”, was rejected as “utopian”. He privately published it in 1923 and expanded the work to, “Ways to Spaceflight”.
In 1929 Oberth became a scientific consultant in Berlin on the first movie to film scenes set in space, “The Woman in the Moon”. His main task was to build and launch a rocket as a publicity stunt before the movie’s premiere. He was teaching at the Technical University of Berlin and his students, one who was Wernher von Braun, helped with the rocket.
And a scene at 1:52:48 seems to clinch that they knew what they were doing--it shows a logbook entry which is translated as "On board everybody is fine. We have reached the distance of 227,000 km from earth. Stopped the last jets. Flying now without jet power. Weightlessness on board." Immediately subsequent to that, the crew is indeed shown as being weightless.