My answer could be late for this question, but I hope it will be helpful for other people looking for similar information.
I had taken a course about Mathematical Logic at National University of Singapore, in which the lecturer used this textbook:
A Concise Introduction to Mathematical Logic, 3rd edition, by Wolfgang Rautenberg
Personally, I like both the textbook and the course a lot.
The textbook initially appears to be quite difficult to read. However, once getting familiar with it, it is much easier to follow since the system of notation is very clear, the content is self-contained and the approach is by starting from foundation, no vague assumption. For example, this book develops the natural deduction calculus and Hilbert calculus, or prove two incompleteness theorems of Kurt Gödel from scratch.