Model Theory, Proof Theory, Set Theory, Recursion Theory: Where to begin? What comes after an introduction to Mathematical Logic? Also, Where would Formal Language Theory stand among the other four branches of Mathematical Logic (listed in the title)? 
 A: It depends on what your goals are. If you have a basic introduction to logic (e.g. Enderton's book or the book by Boolos, Burgess, and Jeffrey), you have the background to learn significant amounts of each of the four areas. 
One thing most general introductions to logic lack is a solid background in basic set theory, particularly ordinals and cardinals. A nice book like Halmos' Naive Set Theory (undergraduate level) or the first couple chapters of Kunen's Set Theory (graduate level) will remedy that. In particular, you need to be relatively comfortable with ordinal and cardinal arithmetic, proofs by transfinite induction, and with the distinction between $2^{\omega}$ vs. $\omega_1$, in order to digest many mid-level results in model theory and proof theory. 
A: Have a look at the Teach Yourself Logic Study Guide which, among other things, has a map of how the elements of the standard math logic curriculum fit together, and gives a lot of reading suggestions to explore. 
http://www.logicmatters.net/tyl
That will reveal that, once you have nailed down the basics of first-order logic and elementary model theory (so you know about the ideas of e.g. a formal language, a formal deductive system, an axiomatized formal theory, completeness, compactness ...), you can branch off in various directions without worrying a great deal about the others, at least at the outset. For just one example: enthusiasts for recursion theory may need to know very little set theory -- and likewise the other way about (as Asaf commented as I was typing this, you can be a serious set-theorist while knowing very little recursion theory.) 
A: I recommend Carnap's "Introduction to Symbolic Logic and its Applications".  It is quite a bit more advanced than a normal 'introductory' text. He introduces simple type theory, and higher-order logic.  It is a good book to transition from basic logic into more advanced topics.
