Reading materials for mathematical logic

Here are a few suggestions (which depending on your background may be more or less useful):

1. Logic and Structure by Dirk van Dalen. I have used this as a textbook when teaching mathematical logic and for that purpose it is decent. Some people find it a bit dry, but at least it covers a large amount of material in a reasonably clear manner.
2. Mathematical Logic by Joseph R. Shoenfield. This book is, I think, regarded by many logicians as being the gold standard text on the subject.
3. A Course in Mathematical Logic by John Bell and Moshe Machover. This is my personal favorite textbook in mathematical logic. (Unfortunately, it's a North Holland book and so is a bit less affordable.)
4. A Course in Mathematical Logic for Mathematicians by Yuri I. Manin (with contributions from Boris Zilber). I think that pretty much anything written by Manin is worth taking seriously and this book is no exception.
5. Notes on Logic and Set Theory by Peter T. Johnstone. This is a delightful little (literally) book on logic which is highly recommended (perhaps in conjunction with one of the other larger books from this list).
6. The Mathematics of Metamathematics by Helena Rasiowa and Roman Sikorski. This is a nice book which gives a lattice theoretic development of mathematical logic. (Difficult to find, but worth a look if your library has a copy.)
7. Introduction to Metamathematics by Stephen C. Kleene. A classic text in mathematical logic which is still a rewarding read.

I hope these (admittedly biased) suggestions are some use!

I'm surprised my favorite introduction to mathematical logic hasn't been mentioned by anyone. It's Robert S.Wolf's A Tour Through Mathematical Logic. Wolf has written a book that is extremely compelling to read. His passion for the subject comes through in every sentence. It reads like a novel on mathematial logic and set theory, complete with detailed historical notes, philosophical insights and lots of problems. The book is practically a meditation on the answer to any frustrated student of logic's question,"Why is this important?" I wholeheartedly recommend it as your starting point before you look at any of the more-depth treatments recommended below. As a follow up, I recommend the classic introduction by my old teacher, Elliott Mendelson, An Introduction To Mathematical Logic, a deep and masterfully written introduction for graduate students. Those would be my recommendations.

Joe Mileti wrote a really nice set of course notes on mathematical logic (approx 20 weeks of lectures). It's a draft for a book titled, I think, "Mathematical Logic for Mathematicians." The course notes are beautifully written (and beautifully delivered, if you've had the chance to see him lecture). He's a very nice guy, and I would suggest contacting him about it.

My memory is a bit hazy about the topics he covered, but we discussed propositional and first order logic, nonstandard analysis, and axiomatic set theory. I also remember that some highlights included connections to graph theory and algebra (I guess this sort of touches upon his research themes).