Boolos, Burges, and Jeffreys' book "Computability and Logic" (I think it's now in its fifth edition?) is by far the best logic book I've ever run into. The first chunk of the book is focused specifically on computability theory, but one can skip right to the presentation of first-order logic, which is absolutely fantastic. The book covers a number of topics which don't tend to appear in basic logic books - modal logic, second-order logic, forcing in arithmetic - but is still a first introduction to the subject. It's wonderfully written, too.
I also quite like Ebbinghaus, Flum, and Thomas' book "Mathematical Logic," but not as much.