I'm a huge fan of Boolos, Burgess, and Jeffreys, "Computability and Logic." The first-order logic part proper starts with the chapter "A precis of first-order logic: syntax," and the book can be begun there without any loss of continuity. (The previous chapters are a lengthy intro to computability theory; I found it helpful to return to those chapters after finishing the chapters on first-order logic.)
Alternatively, Richard Kaye's book "The mathematics of logic" is quite good; it takes you as far as Godel's Completeness Theorems, at which point it's easy to switch to something like Boolos, Burgess, and Jeffreys to cover the incompleteness theorems.
(And, for a nice preview of Godel to give you some motivation, this paper by Rosser (http://philpapers.org/rec/ROSAIE) is a wonderful exposition of the reasoning around Godel's incompleteness theorems, and further results.)