I'd like to propose an introduction to mathematical logic which would lead into an exploration of one of the many related areas, e.g., set theory or computability theory.
Initially, an introduction to formal logic would be ideal, touching on the techniques, philosophy, and historical development of the field. The emphasis could shift to either the philosophical or the technical, depending on the wants of the class. Some focus would necessarily have to be placed on gaining moderate technical proficiency, as an understanding of quantified, formal logic is the core from which other theories spring.
Once such a foundation is secured, other topics could be addressed. These could include (for the more technically inclined) introductions to set theory, computability theory, or model theory, or (for the more philosophically inclined) modal logic, non-classical logic, or the foundations of mathematics. Perhaps an overview of some of these areas and the importance of their main results could lead to a more thorough exploration of one or two of them.
As it seems that aaaaarg.org is closed down, there are plenty of works in the public domain on these areas; I'd even be willing to whip up a textbook if need be.