Graduate Level Mathematical Logic Textbooks I'm currently a first year mathematics graduate student, and am at an institution which does not have any work being done in mathematical logic, or any logicians on the staff. I've taken a mathematical logic course with the textbook by Leary, 'A Friendly Introduction to Mathematical Logic, 2nd. ed.' (I actually took the class with the author at his university), and am comfortable with it's contents. 
I am looking for recommendations of graduate mathematical logic textbooks that would reflect the work and content done in a graduate logic course, so that I may see more advanced model theory and proof theory and get a better feel for whether or not these are topics that I would more enjoy studying.

Disclaimer: I haven't taken a graduate level logic course.
Peter Smith a retired professor, who used to teach logic at the University of Cambridge put up a guide here with a list of books: http://www.logicmatters.net/tyl/
S. C. Kleene's Introduction to Metamathematics got reviewed by Michael Beeson when it got republished and as I recall, Dr. Beeson said that the book still had relevance for graduate students as a starting point.  It is also the first book on Peter Smith's list.  The most recent review of the book on Amazaon also says: "This 1952 book by Stephen Cole Kleene (1909-1994) is essential for anyone who wants to understand mathematical logic at the graduate level."  It looks like it has over 900 citations on CiteSeer (I don't know if that is large for citations in logic), is still getting cited at present. 
