Some personal favourites:
England Made Me: Interestingly enough for a novel written in 1935, about multinationals.
Brighton Rock: A vicious little tale of a small-town thug.
The Power and The Glory: A very, very catholic novel, but also a cracking thriller.
The End of the Affair: Another very catholic novel, but also very different.
The Quiet American: Hard to express exactly how good this one is. Vietnam, spies, and a huge indictment of "decent people".
Monsignor Quixote: Life after fascism, and a ridiculous priest who may, in fact, be a saint.
It's interesting how much stuff from the school pops up in his novels - from the school motto in "Monsignor Quixote" to the route walked from the school in "The Captain and the Enemy".
Minor anecdote: He visited the school once, and I remember seeing him stop at a corner and turn to look directly at me. Only in retrospect do I realise that he may have been looking across the decades at me as he might have been himself (the architecture and uniform hadn't changed much).
If you're looking for a light read then I recommend "Our Man in Havana" as a taster of Greene. It's amusing and less imbued with the claustrophobia of his more serious novels.
Pretty sure he left it out of the screenplay.
I never got around to reading Greene, despite the connection, and now I feel as though it might be an interesting exercise.
Certainly the biography in this piece makes me interested in him; I can see aspects of myself in him. Unrelated to the school, of course.
I do find the suffocating catholic guilt themes a bit depressing (read "The Ministry of Fear" or "Brighton Rock" for prime examples), but they're extraordinary nonetheless.
(Edit) As you probably know, "Greenes" House and "The Greene Field" were named for his father.