Peter Smith's book is great. It's very readable and contains all the details. The problem is that it doesn't leave anything for you to do! If you want to get your hands dirty and work a few things out for yourself, I'd recommend Raymond Smullyan's book Godel's Incompleteness Theorems. It's a bit terse, but very clear and complete, more like what one would expect of a traditional mathematics text. Most importantly, it contains some very well selected exercises at the end of each chapter.
Edit: It costs a fortune on amazon, but if you look around discount places like abebooks you can find it for a fraction of that price.