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Gallier's Logic Book
There are many textbooks that cover propositional and first-order logic, its syntax, semantics, and proof theory.
This Jean Gallier's book is available online and contains a lot of relevant material. You should not assume that you can quickly read through all of it. The most important sections are:
- 2.1 (sets and relations)
- 3.1, 3.2, 3.3 (propositional logic)
- 5.1, 5.2, 5.3 (first-order logic)
- 10.6 (Congruence Closure)
- 6.5 (Interpolation, but see dependencies)