One of my personal favourites is "Narcissus and Goldmund", which sketches the relationship between two young men who represent an "artistic" and a "scientific" way of thinking and life.
“His goal attracts him, because he doesn’t let anything enter his soul which might oppose the goal. This is what Siddhartha has learned among the Samanas. This is what fools call magic and of which they think it would be effected by means of the daemons. Nothing is effected by daemons, there are no daemons. Everyone can perform magic, everyone can reach his goals, if he is able to think, if he is able to wait, if he is able to fast.”
Everyone gives what he has. The soldier gives strength, the merchant goods, the teacher instructions, the farmer rice, the fisherman fish.
Very well and what can you give? What have you learned that you can give (the merchant asks of Siddhartha)
I can think, I can wait, I can fast.
I don't necessarily regard Hesse's work as necessarily being directed towards a 'coming of age' readership; but, I discovered him at a time when I was struggling with my own "private battles of adolescence" and am grateful in finding his work.
I am surprised nobody mentions Hesse's book of short stories which i highly recommend. They are quite different from his novels and offer a different insight into his world. It's the one book that I am going to to take with me wherever I move.
 The Fairy Tales of Herman Hesse
Hesse's philosophy is so comprehensive, there is no trite summary. Perhaps simply viewing life as a series of symphonic movements, with competing harmonies and melodies is as short as one can get.
Yet what remain closest to me from my days of enjoying their writings is just an essay by Herman Hess titled "Thou Shalt not Kill". Hess, the man and his times, seems to unravel its depth and mystery here. (And Mann's short story, "A Man and his Dog" has a similar staying power for me)