A good place to start would be Prof. Walter Kaufmann’s two lectures on Kierkegaard, readily available as MP3’s. Really an excellent introduction.
Prof. Kaufmann stresses that Kierkegaard’s didn’t want his philosophy to be “palatable”; he wanted it to be “absurd”, indeed “outrageous”. This whole thing about Christianity being deemed “irrational”? Kierkegaard absolutely agreed with that – he thought of faith as wholly different from reason.
By the way, this faith Kierkegaard speaks of was only and exclusively Protestantism. Never once in all his writings does he even mention Catholicism, or Calvinism or for that matter Judaism (he does mentions Jews from time and time, always with loathing and contempt).
Throwing in my own two cents, I’ll add that Kierkegaard, more than anyone, ultimately helped to push Denmark away from Christianity. Georg Brandes, the country’s leading atheist, read Kierkegaard during a religious crisis in his youth, during which he “tried” - in his own words - to become a Christian, but found that because of Kierkegaard he simply could not. Why? Precisely because of Kierkegaard’s insistence on absurd faith, irrational faith, faith against all reason. It was something which Brandes (and with him most “modern” Danes) could no longer stomach, and so the march towards atheism began. As Brandes himself put it: “By Kierkegaard, Danish intellectual life was driven to the extreme point where a leap is required - either down into the dark abyss of Christianity, or to the point where freedom beckons.”
On social issues, Kierkegaard was rabidly conservative. He considered the freedoms granted to the Danish people in the new constitution of 1848 an absolute disaster. While the rigid class-system of yore had kept everyone in place (and the king on top of them all), these new freedoms fostered social ambition, which Kierkegaard loathed - as indeed he loathed everything having to with the “masses”.
So, OK, how did a fanatic, “irrational” Christian with rabidly conservative social views become the darling of Sartre, Camus and the like? Why is he even remembered today, while for example Vilhelm Beck is practically forgotten?
This is, in great part, because of the emphasis Kierkegaard puts on anxiety and despair, both of which would become supremely important for much of the philosophy which was to follow (i.e.: Existentialism).
For Kierkegaard, all non-Christians (plus most people who believe themselves to be Christian, but who in fact are not!) live in despair, in a state of untruth, where all the big questions (about God and the meaning of life and such) are ignored, swept under the carpet, or laughed away. The only way out, as Kierkegaard saw it, was to become a Christian.
The Existentialists tweaked this (beyond recognition) to mean that all people, including the Christians, live in a state of godless despair, quite similar to the one Kierkegaard described - people are afraid to deal with the big questions, afraid to confront the absurdity of existence, and so on. But in adopting Kierkegaard’s philosophy of anxiety and despair, they started out by hollowing out its (Christian) core - they thought there was no way out except the one you made for yourself.
My initial two cents, anyways. Now go check out those Kaufmann lectures!