I'd like to understand what Daisy means in the first chapter of The Great Gatsby, when she says she is "sophisticated". Does she mean she is "blasé" (she says she has been everywhere and seen everything and done everything")?
“You see I think everything's terrible anyhow,” she went on in a convinced way. “Everybody thinks so—the most advanced people. And I know. I've been everywhere and seen everything and done everything.” Her eyes flashed around her in a defiant way, rather like Tom's, and she laughed with thrilling scorn. “Sophisticated—God, I'm sophisticated!”
This is a moment of irony, though because there are possibly several layers of meaning the irony is a bit difficult to sort out.
One meaning is clear in Daisy's claim to be sophisticated: She feels that the idea of being sophisticated is somehow a false and boring notion.
While rejecting the value of being sophisticated, she also claims sophistication. She seems to be expressing a disdain for being sophisticated, but as she gives Nick something of a wink after this speech Daisy is also trying to claim that she is sophisticated.
To this end, she is expressing the idea that she is blase, as you suggest. Life is blase. Yet the excitement that constantly plays in her voice (and, here, in what Nick recognizes as her performance) contradicts her subtle claims of boredom.
Excitement is an integral part of Daisy's character. It is this that constitutes her allure and, to some extent, her danger. Her attitude regarding sophistication is telling in the way she places herself above the concept, subtle espousing her superiority to a concept even as lofty as this one.
Additionally, the reader has been surprised along with Nick to discover that this well-to-do couple, the Buchanan's, invited him to dinner apparently in order to play out a drama regarding their unhappiness. Tom's infidelity is literally featured as dinner conversation and Nick is rather amazed that these are the kind of people occupying the upper echelon's of society.
Earlier, the reader is told that Nick has aspirations for self-improvement. He has brought certain books along with him to the east and plans to become sophisticated himself. However, from his reaction to the dinner at the Buchanan house, Nick clearly wants little or nothing to do with their sort of "sophistication".
Nick listens, somewhat astounded but understanding the subtext, as Daisy goes on talking on the veranda. This speech extends to Daisy's proclamation about her daughter as well. There she says that she hopes her daughter grows up to be a fool.
This is contrary to the general conception of sophistication, of course, and underscores the irony of Daisy's claim on sophistication. Nick's take on the episode helps us to see that Daisy is playing a part (a distasteful one) in order to entertain herself while also demonstrating her self-identified qualities.