I've wanted to understand Gödel's incompleteness theorems for quite some time, but I've always found this find of math to be a bit of a stretch for my background. That is, until I found "Gödel's Proof", which took me from only a passing misunderstanding of the incompleteness to a basic working understanding in just a bit over 100 pages.
The first half of the book mainly covers the notion of consistency and introduces all the background material needed to understand the both the motivation of the proof and the proof itself. The second half of the book focuses on the proof itself, starting with Gödel numbers and mapping mathematical statements onto numbers and quickly moving to the crux of the proof.
While I can't claim to be an expert on Gödel's work, I do feel that I have a decent understanding of the ideas and can move on my next goal of reading and understanding Turing's original "On Computable Numbers, with an Application to the Entscheidungsproblem" which makes extensive use of Gödel's ideas.