"I'm all ears" is a metaphor. The speaker is comparing themselves to a being covered in ears, which are, of course, used for hearing. This emphasizes that the speaker really wants to hear what their dialogue partner has to say, is filled with curiosity, and will listen intently. "I'm all ears" is also a cliché, a figure of speech that has been used so many times it has become worn out and commonplace.
"O heavy lightness, serious vanity" are examples of oxymoron, which is a figure of speech that occurs when two opposites are put together. This creates a paradox or seeming contradiction. How can something be both heavy and light or serious and vain (frivolous, unimportant)? An oxymoron catches our attention and asks us to dig deeper to understand what is going on. Romeo uses these particular oxymorons in act 1, scene 1, of Romeo and Juliet.
"He's from the Big Apple" uses an idiom to describe where the subject of the sentence comes from. An idiom does not provide a literal description but is a phrase that native speakers in a society recognize from living in that society and knowing the language well. In this case, the person in question is not literally from a big apple: he has not been living a giant piece of fruit. Instead, the idiom is widely recognized as a nickname for New York City, referencing its large size and important status in American life.