I don't know your background so I can't say. But here is a free pdf of Shoenfeld's classic text on Mathematical Logic. It's one of the standard grad-level texts on the subject. Have a glance through it so that you can get an idea of what you're in for. This is not an easy book. You shouldn't be intimidated by it, but you shouldn't underestimate the subject either.

https://www.karlin.mff.cuni.cz/~krajicek/shoenfield.pdf

In general I'd say that the prerequisite for graduate study of mathematical logic is a generous amount of what they call mathematical maturity. Meaning that you're comfortable working with proofs. If you've seen undergrad set theory that will help a lot too.

If you're already at a university, ask one of your profs and/or go over to the math department and ask around. Take as many upper division math courses as you can, especially real analysis and abstract algebra. Real analysis is not specifically prerequisite to logic but it will train your mathematical mind like nothing else. Abstract algebra is IMO prerequisite because it teaches you to handle mathematical abstractions that at first glance don't make sense or that seem unmotivated.

Mathematical logic at the graduate level is nothing like the truth tables and so forth that you probably learned in the discrete math class they give to CS majors.

Good luck!

Abstract algebra is IMO prerequisite because it teaches you to handle mathematical abstractions that at first glance don't make sense or that seem unmotivated. — fishfry

I'd say more than that: current research in mathematical logic is for the most part intimately connected with abstract algebra, algebraic geometry, and combinatorics more generally (including especially matroids). In model theory, for instance, most current research focuses on classifying algebraic structures (Shelah's classification program), so if you don't have a thorough understanding of the structures themselves, the discussion is impenetrable. Moreover, many concepts and techniques employed in this area are inspired by their algebraic analogues (e.g. algebraic closure, Zariski geometries and Zariski topology, etc.). So a certain familiarity with them is required. Even areas apparently more remote from algebraic concerns benefit from this background: e.g. there is a lot going on in proof theory that requires a good knowledge of category theory.

Next, I'd also add to the mathematical prerequisites at least some knowledge of topology. It's incredible how much topological notions are employed in basic every area of logic. If you have already taken a course on basic topology (which covered, say, metric spaces, compactness, separation axioms, etc.), I strongly recommend reading Steven Vickers's Topology via Logic. Vickers motivates the traditional topological axioms (a set of open sets must contain the whole underlying set and the empty set, be closed under arbitrary unions and finite intersections) by noting that they are essentially axiomatizing the notion of a semi-decidable property. This is crucial for understanding why "topologizing" things is such a useful technique in logic (and in general!).