Since I am a language teacher (at least in part of my professional role) and an academic specialist in the artistic use of language (i.e. poetry), and a poet and translator myself, I have thought about what it means for someone to be interested in language, and what people mean by this.
One of the main areas of interest is in stigma. People love stigmatizing others for how they talk or use language. Whether it be those interested in the enforcement of "zombie rules" (non rules) of grammar like split infinitives, or in stigmatizing accents, regional dialects, or vernacular habits of speech as ignorant or lower class. At its least offensive, an interest in stigma becomes an interest in mere sociolinguistic difference, without a negative value attached, or an admiration for prestige dialects.
(Disfluency is supposed to be a sign of moral or political corruption. I never understood that. I might hate the linguistic habits of people I agree with politically, for example. Wouldn't that be more my problem than theirs? Or if right-wing people are poor spellers, well, that means that they are poor spellers, and nothing more than that.)
Another category of interest is in words and etymologies. Some people who like language just like word origins. That's fine. I happen to know some etymologies, though I can't say that's the source of my interest in language. I like words, but I like sentences more than words per se.
Some people are interested in the process of language learning. I have some interest in that in my role as teacher.
People are interested in Whorfian ideas about language shaping thought.
But most of my interest in language is in the area of linguistic prosody: rhythm and intonation, pretty much. I think syntax could be pretty interesting, but I don't follow technical discussions of it, lacking formal training. Cognitive linguistics, related to the study of metaphor, is also worth my time, as is the study of idiomatic expressions and proverbs.