First of all, before you read the full proof, it's worth finding a good overview so you know what you're getting into. It's pretty old, but I'm a big fan of Rosser's expository paper on the topic; I'd read that before diving into the full proof, since it clarifies a lot of what will be going on. Note that it doesn't just discuss Godel's incompleteness theorems, but also a couple related results (including Rosser's own improvement of Godel's theorem - which is, in fact, how Godel's theorem is usually stated).
Now what about the full proof?
Peter Smith's book An introduction to Godel's theorems is extremely good, but covers a lot of ground - in particular it doesn't actually get to the incompleteness theorem until chapter 21! So if you're willing to devote a lot of time to this task, I think this is the best source you'll find, but there are definitely faster ways to get there.
If Smith's book seems a bit daunting for now, I recommend the text Computability and logic by Boolos, Burgess, and Jeffrey. I absolutely adore this book; it was the first logic text I read, and it was what got me hooked. As the title indicates, this isn't just about Godel's incompleteness theorems, and so you can ignore most of it for now (but seriously the whole thing is excellent - do read it sometime!) Specifically, chapters $9,10,14,15,16,17$ give an entirely self-contained path to the first incompleteness theorem in about $80$ pages; chapter $18$ then covers the second incompleteness theorem in about $10$ more pages.