This summer, I am directing an independent study on non-classical logics. In part because of Ole's glowing recommendation, the primary text has been Graham Priest's Introduction to Non-Classical Logic. The book has been fantastic; I can recommend it without qualification. It is pitched at just the right level for a philosophy student with maybe one logic course under their belt -- neither too slow nor too quick. The end-of-chapter exercises are also just right: neither too difficult nor too easy. And each chapter closes with a couple of pages dealing with how the technical material presented there connects up with overtly philosophical questions, keeping up motivation for people whose primary interest is not in the formal/ mathematical side of things.
Another aspect of the book that appealed to me was that (partial) soundness and completeness proofs were given at the end of each chapter, separated from the main course of discussion as optional material. Such proofs are of course incredibly important to practicing logicians, but I sometimes think that the amount of time and effort needed for them is better spent elsewhere given the limitations of a classroom, and the fact that most philosophy students in logic classes won't go on to be practicing logicians. The nice thing about Priest's presentation is that if you think soundness and completeness proofs are essential, you can cover them, or if (like me) you'd rather spend that time covering a wider array of logics, you can easily skip over them without loss or inconvenience.
Last but not least, I certainly have learned a thing or two (or ten), even though it is labeled as an introductory textbook. I will definitely use this book again in future classes.