You should read Wittgenstein's blue/brown books. There are a lot of thought experiments that you have to think through, kick the tires on, etc. and he kind of holds your hand at times, while at others you have to do the work on your own. This sort of thing you should read recursively; you're not going to understand the first time you read most of it.

Predicate logic is also something you should study. It relates to the blue/brown books, but it would be incorrect to say the blue/brown books deal with predicate logic, per se (they're concerned with what's "underneath" predicate logic, if that makes sense.) You can take an intro math course at just about any community college/university that will cover the basics of predicate logic (first-order logic is generally considered the basics of predicate logic, but it's kind of a broad domain and this is convention more than anything.) Check out this description for math 414 and find a course close to you that has a similar description. Different schools will put this type of course in different categories, but it will be either math or logic. You shouldn't really need any mathematical knowledge outside of algebra and maybe pre-calc, but unfortunately the school can arbitrarily make the math as easy or difficult as they choose, so the prerequisites will vary and because these courses usually target engineering students they sometimes draw in a lot of material that is unnecessary and non-trivial to learn. A community college will likely have less prerequisites, but you'll have to look at what's available near you.

Unlike the user elsewhere, I don't recommend taking a humanities-based logic course or looking at logical fallacies. These are generally thin on logic and heavy on rhetoric. There are potentially an infinite number of "logical fallacies" and you can't learn them all. Learning formal mathematics will teach you to reason systematically. This is a good place to start if you want to think for yourself.