Hello and welcome. The first thing I notice about your question is that you describe English grammar as if it comes predetermined with labels for things. In fact, grammar is just what internal knowledge we have that allows us to speak-- whether professors or high school dropouts! Linguists are interested in figuring out how all of that works (usually from an abstract, language-based structural perspective, sometimes biologically), although having terms for parts of grammar is of course helpful as we try to describe things and figure out how they fit together. On the other hand, "grammar" as an area of study goes back much earlier to traditional grammatical descriptions, especially of Latin and Greek, where it was thought that there were right and wrong ways to say things (and right and wrong languages). That's what you'll most likely get from a language textbook or your English teacher. That's also where terms like "noun" come from, although we also use them as linguists to describe things.
So from here you have three directions you could go in:
1. Study traditional grammar. Look at the history of grammatical descriptions, perhaps starting with the Greeks, and then up to the 1800s or so with the books when they thought they had it all figured out. They didn't. But it's interesting from a historical perspective.
2.
Prescriptive grammar: the idea that there are standards and 'best' ways of doing things, like taught in an English class. There are times this is very useful, like learning how to write a resumé that will get you hired for a job. When to use commas, and so forth. Also useful when learning new languages (in addition to just practicing, of course).
3.
Descriptive grammar: where linguists try to describe how people speak (rather than telling them how to), and figure out how language works. People speak. But how? We want to know. And you won't really find linguists doing anything prescriptive, because we're not interested in figuring out the "right way", because we don't believe that one language/dialect/expression/whatever is better than another. We're just interested in understanding more about how people can do this complex thing called language.
As for the labels, they're useful for any of the approaches (and at least the major ones are shared across them). But they're really more like myths we tell ourselves in order to have something to talk about. At least until someone really figures out how everything works, there's no sense in which there are "real labels", just various different suggestions of categories and names for them. Borderline cases are fascinating to linguists, annoying for prescriptivists, and often overlooked historically. (That's what I do a lot of my work on, one way or another.)
Oh, and as for a book I'd recommend, here are some ideas:
1. If you like the topics I've mentioned, I'd recommend Anne Curzan's 2014
Fixing English: Prescriptivism and Language History. It's a nice balanced perspective and would introduce various topics that you might want to learn more about later.
2. For a very accessible and enjoyable introduction to the origin/history of language as well as how it is used in society is Tore Janson's 2012
The History of Languages: An Introduction. It's easy to read with no background but gets into lots of interesting questions. At times I disagreed with the author (like what the future of languages will be like in the last chapter) but it's the sort of book that also allows you to form your own opinions while giving the relevant background. Highly recommended. (There's also an older book by the same author called
Speak that might be easier to find at a library and also has a lot of the same content-- the new book seems to be a revised edition of that one to some extent.)
3. Mark Newbrook's 2013
Strange Linguistics is an overview of bizarre (and almost certainly wrong) theories of language. It covers lots of different topics and might point you in the direction of some other questions to explore later, and it's amusing. It is not widely available (even at academic libraries), however. Some of the material is found on a blog here though:
https://skepticalhumanities.com/2013/03/17/linguistics-hall-of-shame-1/There are of course various other books that are more commonly recommended and others would suggest. These are just ones I happen to like and think might be relevant for you.
For anything else please start a new discussion in one of the relevant sub-forums to discuss more, ask for book recommendations, etc. Questions are welcome.
