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).