Models of Peano Arithmetic
The first part of the book gives rigorous proofs of Gödel's incompleteness theorem and related results. Kaye then gets on to the properties of non-standard integers, such as Tennenbaum's theorem which says that arithmetic on non-standard integers is not recursive - i.e. you can't do calculations in the way you are used to with other mathematical systems. You might think that this made it impossible to study the properties of non-standard integers, but that is what Kaye does in the rest of the book, deriving results such as Friedman's theorem - every countable non-standard model is isomorphic to a proper initial segment of itself.
Kaye says that some of the book should be understood by an advanced undergraduate student. I have my doubts about this, but the book is well laid out, and will give a flavour of the subject even to those who can't follow the details. Also it's probably the best book if you want to study non-standard integers in depth.