I have a great idea for a jukebox musical but don't know what copyright laws or permissions are involved.

The answer depends on how you are going to use the music in your musical. Performing rights organizations like ASCAP and BMI can give blanket music licenses for the performance of music, called “small” performing rights. Under small performing rights, you can play or perform a song publicly where the song is not integral to a story. Small performing rights are things like radio airplay or live performances in a restaurant or bar, or even concerts where a variety of music is played without an underlying story. If the song performance is in a dramatic setting, such as a stage play or musical where a character sings a song, you are talking about "grand performing rights". Grand performing rights are controlled by the composer and/or music publisher of the song you want to use, NOT ASCAP and BMI, and you must get permission from each song and negotiate a license. It is a lot of work, but it is much less work than if you write the musical and then find out you can't get a license to a song.