Though not too many records exist of the music that was used during performances of Shakespeare's plays, we have a pretty good idea of what it would have sounded like.
Back in college, I fell in love with “early music” - anything before the Baroque era (so just about anything pre-1600). Part of the blame falls on the music history classes that happened to be offered while I was there, and part of the blame falls on joining the Early Music Ensemble, which you can see in this lovely, doofy picture we took at the end of one semester (with my classmates' faces obscured; clearly my photo editing skills are top-notch).
What made that ensemble so much fun was the instruments, which made everyone an amateur: no one grows up learning how to play the viol (well, basically no one). No one takes krumhorn lessons. This ensemble was, therefore, mostly amateurs to some degree. If you were a cellist, you played viol, but you were also asked to learn a wind instrument. If you were a clarinetist, you picked a wind instrument, but also had to learn a string instrument. Everyone had to sing.
Luckily, I was a Celtic harpist who had played recorder in 2nd grade, so I guess I kind of cheated. At any rate, our concerts provided a bit of a glimpse into what music sounded like in the 16th and early 17th centuries: a little bit of wind band music, some lute, viol, and harp music, and a smattering of madrigals.
If you were kicking around England in the late 1500s and stumbled upon a performance of a Shakespeare play, you might have heard some lute and viol interludes between scenes. A wind band might have preceded the play, welcoming the audience. Dance scenes could have been accompanied by a consort - a mix of any of these instruments (an in-depth look with sources can be found on Encyclopaedia Britannica).
Want to hear some of this music and learn more about the instruments? You’re in luck. Piffaro, Philadelphia’s Renaissance wind band, is (unlike my ensemble) not a bunch of amateurs. They are incredible musicians, and they’ve put together a series of videos that walk you through why a krumhorn sounds the way it does, why Renaissance harps sound so plinky, and what the heck a sackbut is.
Recorder: it’s not just a plastic toy, and there are a whole bunch of sizes.
Loud Band: sackbut, slide trumpets, and more!
Strings: harp and lute, together forever.
Krumhorn: the kazoo of the 16th century.
Since Piffaro is a wind band, they don’t have a video about viols. A viol is the precursor to the modern violin, viola, cello, and bass. There are 6 strings instead of 4, and no matter what size - soprano, alto, tenor, or bass - the player holds the instrument between their knees. There’s no endpin to hold it up, so it’s a little bit of a leg workout too. Here's Phantasm playing a Pavan and Galliard by William Byrd: