A word about the label first: several of the people filed under existentialism rejected it. Camus denied being an existentialist repeatedly, and Heidegger refused the classification outright. What the figures share is a starting point rather than a doctrine — the concrete situation of an individual having to live and choose, rather than an abstract system.
Camus is the best entry because The Myth of Sisyphus states the central problem in a form you can hold in your head. We crave meaning; the universe offers none; the collision between the two is the absurd. His question is what follows, and his answer is neither suicide nor a leap into belief, both of which he treats as evasions — but living in the tension deliberately. The image of Sisyphus, condemned to roll his boulder forever and imagined happy, is the whole argument compressed.
Sartre's public lecture, published as Existentialism Is a Humanism, is the most efficient summary of his position: existence precedes essence, meaning there is no human nature fixed in advance, so we are what we make ourselves. From this he draws the famous claims that we are condemned to be free, and that we are responsible not only for ourselves but for the image of humanity our choices project. Sartre later had reservations about the lecture, which is worth knowing — treat it as an accurate introduction rather than his considered view.
Simone de Beauvoir is the most unjustly skipped. The Ethics of Ambiguity tackles the problem the others leave open: if we are radically free and there are no given values, what stops existentialism from collapsing into "anything goes"? Her answer turns on the fact that my freedom is bound up with others' — willing my own freedom coherently commits me to willing theirs, since a world of oppression constrains everyone's projects. It is the movement's most serious ethical work, and it is directly argued rather than literary.
Kierkegaard, writing decades earlier, is the tradition's ancestor and a difficult starting point because much of his work is pseudonymous and ironic — he stages positions rather than asserting them, and taking a pseudonym's view as Kierkegaard's own is the standard error. Fear and Trembling is the usual entry, on faith and the individual's relation to the universal.
CautionsCommon misreadings.
It isn't pessimism. The recurring claim across these writers is that the absence of given meaning is what makes creating meaning possible. Sartre thought his position was optimistic precisely because it left everything to be done.
"Anything goes" doesn't follow. This is the objection existentialists worked hardest on. De Beauvoir's answer, and Sartre's appeal to responsibility for the image of humanity, are attempts to show that radical freedom generates constraints rather than removing them.
Don't start with the treatises. Being and Nothingness and Being and Time are major works written in dense technical vocabularies. Read cold, they produce the impression that the movement is deliberately obscure — which is exactly the impression the essays and novels disprove.