Just as in any area of mathematics, different parts of the subject have different strengths. The theory of Reverse Mathematics investigates these strengths by determining for a large number of mathematical theorems, including theorems arising in the development of first order logic, the weakest axioms that are able to prove them. In order to do this, one reverses the usual direction of mathematical proof, by proving the axioms from the theorems(, over an extremely weak base theory)theory.
The remarkable fact of Reverse Mathematics is that many/most many theorems of classical mathematics fall into five large equivalence classes, consisting of provably equivalent theorems. Many instances of this are listed on the Wikipedia page I link to above. I believe
It appears that all the most trivial parts of logical syntax can be developed in the weak base system $RCA_0$. Beyond RCA_0$, and this , here are some may be the answer to your question. For more powerful parts of the less trivial examplestheory, one needs stronger axioms.
A weak version of the Goedel Incompleteness Theorem (for countable theory, already closed under conseqeunce) is also provable already in the base theory $RCA_0$. (As is the Baire Category theorem and the existence, but not uniqueness, of an algebraic closure for any countable field.)
The Completeness Theorem for countable languages is equivalent to $WKL_0$. (As is the Heine-Borel theorem, the Jordan curve theorem, and many others.)
the sequential completeness of the reals and various forms of Ramsey's theorem are equivalent to $ACA_0$.
Comparability of well-orders and Open Determinacy are equivalent to $ATR_0$.
the Cantor-Bendixon theorem is equivalent to $\Pi^1_1-CA_0$.
