There are certainly many different aspects you could point to. However, one of my favourites is seeing how Nick is different when he sees Tom Buchanan for the last time at the end of the novel. Although initially he is very cold with him, he suddenly has almost an epiphany concerning them and their natural propensity to destroy things. He says, famously:
They were careless people, Tom and Daisy--they smashed up things and creatures and then retreated back itno their money or their vast carelessness, or whatever it was that kept them together, and let other people clean up the mess they had made...
Nick, after this realisation, decides to shake hands with Tom, because he felt as though he were "talking to a child." Thus part of Nick's coming of age is an accurate understanding of upper-class characters like Tom and Nick, and how they operate and the chaos they leave in their wake. Instead of being enchanted by such riches and glamour, Nick now has a very accurate idea of what their characters really consist of, and is no longer dazzled by them.