Kaye, R., 'Mathematics of Logic' is a good first-year text. Also consider Boolos 'Computability and Logic', but this could get in the way if you have a particular way of teaching CS/computability topics.
I prefer these to the Mendelson - which I found a bit confusing for the sake of formal accuracy. Kaye, by example, avoids too much technical jargon, and keeps to the ideas in play, building to a completeness theorem.
Hope this helps!