New to logic, what should I study first? What are the different types of logic, like formal logic, propositional logic, predicate logic, first order logic, etc and how are they the same and how are they different. And if I was going to start studying one of them, what should I start with?
 A: There are a ton of introductory books on mathematical logic; I will mention just a couple, and then attempt to answer your question.

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*Introoduction to Mathematical Logic, by Elliott Mendelson

*A Course on Mathematical Logic, by S.M. Srivastava

"Formal logic" and "mathematical logic" mean pretty much the same. A very large and important part of this is the study of so-called first-order theories. As an example of a statement of a first-order theory, consider this, from the first-order theory of fields:
$$\forall x[\neg (x=0)\rightarrow\exists y(x\cdot y=1)]$$
which means:

for all $x$, if $x\neq 0$ then there exists a $y$ such that $x\cdot y=1$

The symbols $\neg$ and $\rightarrow$ are called propositional connectives; they stand for "not" and "implies". Other such connectives are $\wedge$ and $\vee$, standing for "and" and "or". You can study the use and meaning of these connectives apart from the other aspects of first-order logic. For example, the part inside the brackets of the above statement has the form $\neg A\rightarrow B$.
Propositional logic is this study; it's the most basic layer of formal logic (also the oldest historically). Here's an example: $\neg(A\wedge B)$ is true precisely when $(\neg A)\vee(\neg B)$ is true, regardless of what assertions $A$ and $B$ stand for.
Things start to get more interesting when we add the quantifiers $\forall$ and $\exists$ ("for all" and "there exists") into the mix, along with symbols for relations (like $\lt$, =), properties (also called predicates), constants (like 0 and 1), and functions and operators (+, $\cdot$). A first-order language is what results. The particular collection of relations, functions, etc. will depend on the subject area. Add in axioms written in this language, and you have a first-order theory. The first-order theory of fields, for example, uses the constants 0 and 1 and the operator symbols + and $\cdot$.
Some stuff is true across all first-order theories. Example: $\neg\forall x(A(x))$ is true precisely when $\exists x(\neg A(x))$ is. Here $A(x)$ is some property of $x$ (e.g., $x=0$). The study of this universally true stuff is called predicate logic.
(In practice, "first-order logic" and "predicate logic" mean nearly the same thing. It's not worth spending too much time trying to draw distinctions between these terms.)
There's another dimension to all this. The study of formal logic can be divided into syntax, semantics, and proof theory. Syntax refers to the precise rules for forming statements in the formal language. For example, $\neg A\vee B$ is a well-formed expression in the propositional calculus, $A\neg B\vee$ is not. This is the most basic aspect of formal logic (and the most boring, IMO, but like practicing scales in music, it's essential).
Semantics refers to the meaning of the expressions of the formal language. For propositional logic, truth-tables can be used to specify the semantics. For predicate logic, it's a more complicated story.
Finally, the term proof theory is, I hope, suggestive on its own without further explanation. But a great deal of the non-obvious lies behind those innocent two words.
Almost any introductory textbook will begin with syntax and proposition logic, then move onto predicate logic and first-order theories. After that it may consider particular first-order theories of special importance in the history of logic: number theory and set theory. Gödel's famous incompleteness theorem, for example, is a result about first-order number theory. (As it turns out, this theorem also applies more widely.) Mendelson's text follows this route.
One more thing: the examples I gave may seem rather boring. I haven't said anything about what makes mathematical logic interesting. Had I written about that, I wouldn't have addressed your question. It's as if you posted a question about the French language, and I discussed conjugations and parts of speech, but never mentioned Flaubert or Proust.
