Most books on logic are aimed at pure mathematicians. Here is a lovely overview of lots of "applied logics", directed at computer scientists. It covers a range of uses of logic: propositional and predicate logics, conventional program proving, engineering a modal logic for your particular application, and model checking.
Everything is motivated by good introductory sections and good examples, before leaping into the symbols. One nice theme running through the book is that of comparing and contrasting doing proofs by using inference rules, and finding meanings by calculating the semantics.
Given the breadth, nothing is covered in sufficient depth to be able to go out and start using the techniques in anger. But it gives a solid grounding, ready for the next stage of learning, and the uniform treatment of such a wide range of logic applications gives a wonderful feel for the underlying unity of the subject.