I gave a course in logic using the lecture notes "Sets, Models and Proofs" by Moerdijk and van Oosten, and the course went quite well. The text is very well written and can be recommended for self-studying. The book does not start with first order formula, but with set theory. I think this is an advantage, because the formal arguments involving the syntax of first order logic are considered boring or confusing or both by many students.
Computability theory is missing, as is incompleteness, so unless you want to teach logic as an introduction to model theory you have to take some other text for these parts.
The notes are freely available via the authors homepage http://www.staff.science.uu.nl/~ooste110/onderwijs.html .