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), optionally Algebras Appendix
• 3.1, 3.2, 3.3 (propositional logic)
• 5.1, 5.2, 5.3 (first-order logic)
• 5.4 (proof system for first-order logic)
• 6.5 (interpolation)
• 8 (resolution)
• 10.6 (congruence closure algorithm)

However, please check the dependencies for chapters.