Stanza 113 provides a good example. Harold (in soliloquy, so to speak) discusses how he has not "loved the world, nor the world [him]." The reason is that he has refused to bow to social conventions, to do what the world expects of him. This has caused him unhappiness--he calls his thoughts "a shroud"--but he cannot help being what he is. In the midst of his melancholy, there is the hope that he'll achieve happiness at some point, but it is an individualistic happiness that would seem to be impossible on this earth. At best, one might achieve a sort of state of nature existence as in Rousseau.
The sort of individualism that separates one from almost all of one's fellow men--it's not accidental Harold's daughter is mentioned at the end of the selection and that he'll never get to be near her--is a hallmark of this type of Romanticism. Harold is the first example of the dark, brooding, mysterious, and dangerous hero that came to be called "Byronic."
In Canto II, Childe Harold expresses his preference to wander alone, away from the general throng of humankind. The importance of the individual separating from the mass of humanity is a typically Romantic posture. Harold seeks out the solitude of nature and in these passages contrasts it to crowds, which he calls the "the shock of men." He says he would rather be a lonely "eremite," or hermit, on an island than in the midst of people.
Childe Harold is typically Romantic in seeking out and responding to nature. He feels more alive and himself on a lonely island, amid "waves so blue," than around people. Here he can be himself. He finds nature holy, or "hallowed."
The quintessential Byronic hero, Childe Harold carries with him a sense of injury and pain. He does not find himself through community or human interaction but by relentlessly seeking his own path. He "hate[s] a world he had almost forgot."