In my experience, both with myself and watching others, symbolic logic doesn't help except in computer science and a little in math. Usually when people learn symbolic logic, they acquire skill at playing a game with symbols, something like playing checkers or WFF ’N PROOF, but no skill at understanding concepts, propositions, or reasoning. If anything, symbolic logic hurts, because it uses the vocabulary that ordinarily describes thought to describe the meaningless symbol game, making it harder to talk about the former. Many people have told me that after they took a course in symbolic logic in college, they did not see what it had to do with reasoning, nor did they see any use in it, and they quickly forgot the rules for fiddling with the symbols. Outside of computer science, almost no one uses symbolic logic for anything except symbolic logic. It's just too hard, too abstract, too far removed from the rest of the world.
Traditional logic has its faults, but it provides straightforward and intuitive vocabulary for talking about stuff like what sentences mean. Traditional logic is never far from traditional grammar, which is mostly concerned to identify how logical relationships are expressed in language. (Modern grammar is primarily concerned with syntax: rules to distinguish legal from illegal ways to combine words into sentences—not what you need if you want to gain skill at understanding what sentences mean.)
Unfortunately, I don't know of a good book to strongly recommend to learn traditional logic. I picked it up from a variety of sources. One of those was Raymond McCall’s book Basic Logic, which is decent, though rather dry, has so-so examples, and occasionally has an axe to grind. I've only read a little of the Port-Royal Logic, but what I read was absolutely delightful: the examples came from poetry, history, wisdom, and ordinary life. The prose was playful, intelligent, and unabashedly opinionated. The authors were concerned to elucidate thought, not merely to play games or split hairs. I'd say the Port-Royal Logic is probably the best source to look at first.
Watch out for books on traditional logic that emphasize the syllogism. Some of them treat memorizing valid forms of the syllogism as the main point of logic: learning to tell valid from invalid deductive reasoning—by, of all things, memorizing lists of syllogistic forms. That's useless. The most important thing is to learn the conceptual vocabulary and a few principles. The rest is triviality.
I can tell you some of the main, useful ideas of traditional logic right here.
A proposition is the kind of thing that's true or false: an affirmation or denial that something is a certain way—that some abstract quality or relationship belongs to some indicated subject. Propositions usually correspond to sentences in language. For example, the words "My cat Jerome usually checks out the contents of the refrigerator whenever I open the door" expresses a proposition about my cat Jerome and his usual behavior when I open the refrigerator door. It's true if he does that, and false if he doesn't.
A term is one of the elements of a proposition, like my cat Jerome, his behavior, the refrigerator, etc. Terms are not true or false. You can verify that to yourself by using the word or words for one without making a complete sentence. For example, "Jerome". See? Not true or false.
A proposition has a subject: what it claims something about. And a proposition has a predicate: what it says about it—what is affirmed or denied of the subject. Predicates are always abstractions, like qualities, relations, actions in the abstract, etc.; subjects can be concrete or abstract. In a simple example, like "Jerome is fat", the subject is "Jerome" and the predicate is "fat". Notice that in language, "Jerome" is a proper noun: a name for a specific thing; and "fat" is a common noun, a name for an abstraction. In more-complex propositions, you have some flexibility in what you call the subject and what you call the predicate. In the example above, you could say that the subject is Jerome's usual behavior around the refrigerator door when I open it, and the predicate is checking out the contents. Or you could say that the subject is just Jerome, and the predicate is everything else. It doesn't matter a whole lot; what matters is that you understand what is being affirmed or denied: that you understand what is about talked about and what is being said about it. If you can understand that, you understand the literal meaning of a sentence that expresses it.
Regarding negation, there are two kinds: denial, in which a whole proposition is negated, like "Jerome doesn't poke his nose in the refrigerator", and negation of a term or part of a term: "Jerome quickly becomes bored with things that he can't eat." That he can't eat restricts things, so the proposition doesn't say anything about whether Jerome quickly becomes bored with things he can eat. A negative proposition makes a claim: that its predicate does not belong to its subject—and so a negative proposition is true or false, just like an affirmative proposition. Negation of a term or part of a term restricts the scope of the term, and isn't true or false. "Things that he can't eat." See? Not true or false.
Negation gets complicated when you try to follow the implications of negating something on what's inside it. For example, think of what happens when you negate the proposition "Jerome doesn't care about things that he can't eat." That would be: "Jerome does care about…um…things that he can eat?" Nope, it doesn't work like that. To affirm what the first proposition denies, you just say "Jerome does care about things he can't eat." Negation gets complicated when you consider something called quantification: whether you are referring to all, some, none, just one, most, a few, etc. of something, and whether you are saying that an instance exists or not. Symbolic logic shines at this, but it follows a convention that's mostly suitable for math; traditional logic follows a convention that's more suitable for ordinary matters (and terrible for math). But basically, the negation of "At least once, Jerome has eaten food that doesn't belong to him" is "Jerome has never eaten food that doesn't belong to him." This is one place where the subject can get legitimately complicated—more than I can explain even in a long message like this.
You don't need a "calculus", and you should not treat traditional logic as a calculus. People often speak of translating sentences "into logic", then manipulating them according to the rules of logic, and then translating back. The standard criticism of traditional logic is that, when treated as a calculus, it can't determine the validity of certain kinds of deductive inference. Of course, the same is true of symbolic logic; it can't substitute for reasoning, either.
Some writings on traditional logic treat it as a calculus for deductive reasoning, just like symbolic logic. I recommend ignoring that. Traditional logic is not a calculus, and does not presume to distinguish all valid inferences from all invalid inferences. The main value of traditional logic is to give you some vocabulary to help distinguish important aspects of what people mean when they say sentences. If you understand the difference between subject and predicate, then you've learned to focus your attention on some aspects of sentences that are genuinely useful to distinguish. These concepts focus on the substance of what is meant, not on nit-picky technicalities of wording or which jargon term to use to talk about it. If you just ask yourself, "What is this person talking about, and what are they saying about it?," that alone is often fruitful when sorting out confusion. And that's plenty.
There are some more useful things to know in logic, traditional and otherwise, but if you understand that much, I'd say you're ready to declare victory.
This all said, once you have some basic vocabulary for talking about propositions, it's time to stop studying logic and get back to talking, listening, reading, and writing. That's mainly how you learn to communicate through language: by communicating through language, not by memorizing rules about it or learning words to talk about it. A little bit of "meta" goes a long way; a lot of "meta" just gets you muddled.*
There are many aspects of what is expressed in language that are not covered well by any theory of logic that I've ever heard of. For example, emphasis in language communicates something important, but most logicians take no interest in it. The tense system of English requires that you understand temporal relationships in a certain peculiar way. The past tense of English often indicates hypothetical propositions—imagined to become true in the future, even. Logic can give you some helpful vocabulary for making sense of this, but ultimately the details of language must be understood subconsciously, through experience. They're too complex to fully understand consciously.
* If you want to push further, and you're willing to accept getting muddled, then please don't let this message discourage you. Often getting muddled is a step toward learning something valuable, which you couldn't even imagine before you spent a long time muddled. But then you shouldn't think of it as a means to an end known in advance, you should think of it as indulging your curiosity and following it wherever it leads.