I used to teach Existentialism before leaving academia. As the other comment mentions Kierkegaard's "Either/Or" is worth a read, I also recommend "Fear and Trembling."

The other two writers associated with this term I think are must reads are Dostoevsky and Nietzsche. The former's Notes From Underground is one of my favorite pieces of literature. I highly recommend reading it alongside his "Winter Notes on Summer Impressions," a travel diary he wrote that heavily inspired his later work. In it, he vividly describes the kind of anxieties he has with modernity that influences his work.

And for those wanting a higher level overview of Existentialism and the myriad (seemingly incompatible) writers associated with the term, I highly recommend Kevin Aho's "Existentialism: an Introduction." I've read most of the popular overviews and this one is hands down the best.