My favourite introductory book on mathematical logic is Robert S. Wolfe's A Tour Through Mathematical Logic. Amazingly well written, it covers an extraordinary amount of material in both logic and set theory, complete with biographical vignettes and historical insights. There a deep discussion of first order logic, and it's place in metamathematical systems. It also has a wonderful introductory chapter on the theory of computation and its origins. This is the book I would recommend to any of my students if they asked me about logic.
After that, there are several more advanced texts that I like, particularly the ones by Shoenfield and the classic by my old teacher, Elliot Mendelson.
You can't go wrong with any of those.
