I am writing my bachelor thesis in english, but I have studied logic in my different native language. Since a rely on elementary results from first-order predicate logic (especially satisfaction in structures), I should reference the reader to a textbook for details.
Is there a standard textbook that starts with elementary propositional logic and goes to Incompleteness theorems?
Hamilton's Logic for Mathematicians might be a good choice. The book is elementary and it also covers all that you said. In addition, the book is not very wordy or involved in tangential issues, which means that it might be easy to use as a reference for words and phrases you want to find.