I'm not a big swing fan (more of a hard bop head), but The Benny Goodman Story is amazing. There are two reasons that I am going to recommend it.
First, the story. It tells of Goodman's journey the Chicago ghetto to Carnegie Hall. Steve Allen plays the adult Goodman, who is written as a kind of musical nerd. He barely notices his future wife (a stylish Donna Reed), even after having two identical conversations with her. He doesn't care if his bandmates are black or white, all he cares about is the music.
It isn't made that explicit, but the movie also shows the rise of Swing. Goodman played a style of music he called "hot", which wasn't too popular (for most of the movie). His audiences preferred most sedate dance music ("the standard arrangements"). But when Benny finally found his audience (west coast kids who heard is late night radio show in an earlier time zone), they started jitterbugging.
Which brings us around to the other cool thing about this movie: The music and the musicians. The clarinet parts are all really Goodman, with Steve Allen accurately miming. And many of his band members are played by themselves. We get to see and hear Kid Ory, Lionel Hampton, Gene Krupa, Harry James, Ziggy Elman (demonstrating his famous klezmer solo in Carnegie Hall) and even Dave Brubeck. The movie is filled with music played by the founding musicians.
In conclusion, John Hammond, whose sister Alice (played by Donna Reed) married Goodman, is the rich kid who discovers and promotes Goodman. I knew him as the man who discovered and promoted Bob Dylan. Turns out he made lots of people famous, from Billy Holiday and Count Basie to Bruce Springsteen and Stevie Ray Vaughn. He might be my new hero.