You will not find a single general introduction to syntax that covers everything you might come across. This is because of both depth and breadth, mostly the latter.
There are a variety of different syntactic theories out there, and to some extent they are completely different from one another. When students take an intro class, the instructor selects a textbook and teaches them one approach, usually just based on their preference. There are broadly different approaches, as well as more subtle variation within those approaches. Which ones you encounter will depend on the kinds of research you read, the kind of syntax popular at your university, etc. (Ideally, some of the variation would simply be terminological/notational variants, and to some degree that is true, but not completely. Personally I would also add that I think the different theories are not in as direct competition to "solve syntax" as they are often presented to be, but instead focused on answering different, if related, questions.)
So, all of that said, there are various ways to approach this, but you'll first need to pick a framework. Reading material online (e.g. Wikipedia) or in an intro textbook will require this, because otherwise there's just so much to pick from. Probably the most typical approach would be to start with Generative Grammar (i.e. Chomsky's work), and a good place to start there is with the Government and Binding version of that from the 1980s. Many intro textbooks begin there. These days, research in Minimalism builds on that and a basic understanding is something of a prerequisite. There are also various other different approaches: other Generative variants like LFG and HPSG, etc., completely different approaches like Construction Grammar. You can find intro textbooks for all of these. Generally speaking the terminology of the Generative approaches will be harder to understand, so in a sense it might make sense to start there in order to have that foundation, even if you end up going in another direction later. I think all of these approaches are valuable (sometimes for different reasons), and ideally they would all lead us to a better, combined theory in the end that takes insights from across the different perspectives, but we're not there yet. Another approach to consider would be what is sometimes called "Basic Linguistic Theory" (or similar), just focusing on descriptive information and looking at how languages vary (e.g. typology), which is also assumed (at least the terminology) within other approaches when attempting theoretical explanations of phenomena.
To emphasize, I am not exaggerating the differences in theories and what you might encounter in one class or one textbook. Students who have taken an intro class with one approach might be entirely unable to follow even the basic arguments of another approach, and certainly wouldn't be familiar with the notations. (Even a typical Syntax professor may not be able to follow the details, or perhaps even basic ideas, in a radically different theory, at least not without doing some background reading first.) As a basic example, most Generative approaches (though not necessarily all) will involve syntax trees, but some other (e.g. Construction-based) approaches won't have any trees. Just one difference regarding notation, but relevant for how this differs from a practical perspective.
Some specific suggestions, not a comprehensive list:
--Andrew Carnie's
Syntax: A Generative Introduction: a go-to textbook (probably graduate level) for a relatively standard approach to Generative Syntax (including some later chapters on other related and newer approaches)
--Sportiche et al.'s
An Introduction to Syntactic Analysis and Theory, which I personally use to teach intro classes, and is a bit more streamlined than Carnie's larger volume, and for that reason may be more appropriate for complete beginners including undergrads. It doesn't go into as much technical detail, and it focuses more on building a specific theory, while Carnie gives more historical context and extensions
--Koeneman & Zeijlstra's 2017
Introducing syntax, which is an interesting new attempt to start directly with Minimalism in an introductory course
--There are also various (including some very new) good overviews of Construction Grammar, which I have not used to teach so I won't recommend here, but in general it should not be hard to find an overview
--Similarly you can find intro textbooks to LFG, HPSG, and various other approaches
--I could recommend some books on typology, descriptive morphosyntax, etc., if you'd like, but that's going in a different direction.
As a starting point, I might most recommend the following:
https://langsci-press.org/catalog/book/255Müller's
Grammatical theory is an open access book that is not necessarily appropriate as a textbook for an introductory class, simply because it is relatively technical and so comprehensive. It is also comparative, rather than intending to teach one approach or another. But that sounds like it might be exactly what you're after, or at least could help you pick a particular approach to read about more, and since it's open access it won't hurt to look. It has very good metatheoretical overviews of each of the major theories, and is somewhat unique in that way. It's more of a reference work than a textbook, but good for someone generally familiar with ideas in syntax to either review or to familiarize themselves with a new approach. It is particularly useful for understanding the mechanics of different theories, but it is not a light, introductory approach, but rather a technical comparison.
