Where should I start when learning logic? Should I start with syllogistic logic, predicate logic, etc?
In general, one starts with propositional logic. It is a 2-valued logic. Every statement can be proved with the help of truth tables. Propositional logic is also the logic we use in our every-day reasoning.
The study of propositional logic was initiated by Aristole, e.g. in his books of metaphysics.
You can learn propositional logic from any elementary textbook of logic or introduction into logic.
The next step could be either predicate calculus or modal logic.
In standard logic courses, students start with simple propositional logic in order to familiarise with logical operators ("and", "or", " not", "if then"...) that connect propositions. They learn deduction rules for the operators and the truth tables associated with them. Then they go on and analyze propositions in terms of predicates, constants, variables and quantifiers ("for all", "there exists"): this is predicate logic. They learn the deduction rules for quantifiers and sometimes they learn to use Venn diagrams. Only then they might learn other logics (modal, intuitionist, many valued...) or other related aspects (proof theory, model theory...).
In mathematics curriculum, logic is often introduced through Boolean logic, which is just the algebra that corresponds to propositional logic. Syllogistic is interesting for an historical perspective on the development of logic from antiquity. It is the ancestor of modern predicate logic.
I would start with first order predicate logic, which is foundational to others. Then move on to non-classical logics and modal logics.
If you want some book recommendations, I suggest Mendelson's Introduction to Mathematical Logic, Priest's Introduction to Non-Classical Logics, and Hughes and Cresswell's New Introduction to Modal Logic.