In The Fault in Our Stars, what are some interesting parts from the first chapters that are worth studying for language and style?
When you ask about "language and style," you are asking about diction and techniques like, for example, imagery and irony. Diction encompasses the mode of word selection and the vocabulary of word selection.
There are four modes, or levels, of diction: formal, informal, colloquial, and slang. These modes also encompass syntax and grammar since certain grammatical and syntactical forms are expected in formal, for example, while others are expected in slang. Contemporary writers might mix these modes, or levels, for effect, though earlier writers' individual works were more consistently within one mode of diction, with the exception of dialect being introduced in dialogue.
Vocabulary is the specific word choices within the level of diction used. An oft cited example of the variability of vocabulary is Ronald Firbank's use of "killing" in relation to stairs: "a small house with killing stairs," which combines colloquial "killing" with informal narration. The combination of mode of diction and vocabulary contribute to authorial style along with literary techniques.
Style and diction are always established from the beginning of a work, from the first words. Thus study in early chapters should always start at the beginning. Green's opening paragraph makes an interesting part to study. In it, you'll see the introduction of archaic language, which builds the tone of the first-person narrator, and you'll have you're first introduction to the narrator's voice: "Late in the winter of my seventeenth year ...."
It is hoped that the voice offered at the beginning, when Green reveals no details apart from voice, will match the details about the narrator later offered. As it happens, what is heard in voice does not match what is eventually revealed in detail. Thus the passage giving details about the narrator is important to study. Anne Brontë suffered the same problem in The Tenant of Wildfell Hall. Readers hear a woman's voice then are told the first-person narrator is a man. The reverse is true in Fault in Our Stars: we hear a man's voice and later are told the first-person narrator is a woman. This represents a significant failing in style and language.
my mother decided I was depressed, presumably because I rarely left the house, spent quite a lot of time in bed, read the same book over and over, ate infrequently,...
I'm Hazel, I'd say when they'd get to me. Sixteen.
Another early part to study is the conclusion of the narrator's discussion of Patrick. Here you will find the ironic sarcasm that drives the narrator's tone and a good example of the neologism "Hazel" enjoys (neologism: the creation, or coining, of new words). Irony and sarcasm differ in that verbal irony points out the contradiction between reality and expectation, e.g., a support group for fighting depression that depresses, with an intention to find the associated humor or amusement, while sarcasm points out the contradiction with the express intention of hurting or insulting someone, some idea, some place, something, e.g., Patrick and Regular Doctor Jim. Green's style combines irony and sarcasm to undercut the humor that he implicitly acknowledges in the situations he describes.
by exploiting his cancertatic past, slowly working his way toward a master's degree that will not improve his career prospects, waiting, as we all do, for the sword of Damocles to give him the relief that he escaped lo those many years ago.