"Introduction to Mathematical Logic" by Ebbinghaus, Flum and Thomas
Careful introduction, addresses many doubts that one might have about why one does logic in this way and not some other, e.g. whether one is doing something circular when formulating set theory in 1st order logic, or e.g. it proves Lindstroem's Theorem, that says that classical 1st order logic has the highest power of expressability among the logics with completeness and Loewenheim-Skolem.
