I recently gave an introduction to category theory at the Institute of Cybernetics. This crash course introduced several of the basic categorical notions that are of interest from a computer science and logic perspective, while assuming no previous knowledge about category theory. Everything necessary to understand my upcoming talk about categorical models of modal logics was covered.
The slides of the crash course are now online. If you have questions regarding the material, please ask; for example, via a comment to this blog post.