Your first step would be to learn about predicate calculus (first order calculus or propositional calculus with the addition of 'for all' and 'there exist' quantifiers). This will then lead into the major subcategories:
- proof theory (syntactic manipulation of proofs, proof calculi, ordinals)
- model theory (models and structures of other areas of mathematics)
- set theory (axioms of sets, ZF, the axiom of choice, the general continuum hypothesis)
- recursion theory (decidability, recursive functions, lambda calculus, with connections to theoretical computer science subjects like computational complexity)
Of course, as with any body of knowledge, these categories are not mutually exclusive.
These are specialty areas, meaning that research is done under one of these headings. To get into one of these areas, there are many 'elementary' concepts leading in that you probably would want familiarity with. An accessible topic like Goedel's theorems (the completeness theorem and two incompleteness theorems (these are not the same 'kind' of completeness)) will help get some of these necessary concepts.
Also a good area to be familiar with is philosophy of math and foundations; though not exactly mathematics or logic, it informs a lot of the motivations of study of the above categories.
Logic that is not mathematically based can be pursued in philosophy (human language arguments, deductive and inductive reasoning, vagueness), mostly in the area of epistemology.
Your mathematics department may not have advanced logic, but other departments may (though not necessarily by name. Consider looking in the philosophy department (some mathematical logicians get hired there) or the computer science department (the research may be more applied but they may give courses that are theoretical/basic math).
Executive summary: start with predicate calculus and Goedel's theorem.