My favorite 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 it's 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.