Anyone who wants to trace this proof is free to do so. Though the formal logic can be formidable, and must be learned before tackling the proof, only the basic structure is necessary and it is not difficult to learn. It is also necessary to know a little about prime numbers, specifically that every composite number can be decomposed into some unique group of prime factors. Otherwise, all the technical aspects of the proof (barring the conclusion of theorem VI and a bit of the recursion) can be perfectly understood by someone outside of the world of formal mathmatics.
The proof itself is meant for a professional mathmatician. If you are interested and willing this will not dissuade you. To say Godel was not a master of exposition is misleading for he is ,if nothing else, just that. I have heard working through the proof compared to a mystical experience and the proof itself to a symphony. It is truly beautiful to even the mere math enthusiast. Godel is not, however, a college professor and does not wish to explain what need not be explained. This will not be of much consolation when he prefaces a statement with, "of course," for the twentieth time and you have no idea what he is talking about. But if you are not afraid to go ahead when you have tried and failed to understand, and are not afraid to return when you have gained some small piece of the puzzle and try again, everything will come clear. This is the original. All the commentaries are great, and some are even helpful before you get to the conclusion, but they are not the proof and should not be taken as a substitute. They do not suffice the way a generic drug does. There is no way to understand the full scope of the proof if you are not willing to immerse yourself in it and the language it uses. Everything in it is self-referential, you miss the reference when you skip the proof.
Don't worry if it seems to be going nowhere, because you'll get there soon enough yourself and it turns out everything matters(although nothing has meaning). If you want you can skip everything after theorem VI is proved up to the beginning of theorem XI. Then you will have everything you read about.
Perhaps, if you can, you should get a couple other people to work through it with; different perspectives make all the difference, even in math. It's nice to have someone to share your frustration with and sometimes to have it relieved. Plus the delight of watching the theorem build like a wave and crash down upon itself is best shared with others.
Last, and particularly directed toward anyone at DOVER: The proof as you have printed it is horribly mangled! There are countless misprints (more than fifty). This is bad enough in prose but absolutely disgusting in a math text. The second to last line of theorem VI (the proof of undecidability) has a misprint to the effect of changing a negative statement into a positive one. The proof is hard enough as it is. There is a much better translation (without any misprints) in "Frege and Godel" and "From Frege to Godel," which are sadly out of print but may be found at a library. The DOVER is cheap and it reads that way.
If you're interested, do it, and don't worry about it being too hard. You will realize the technical aspects are almost all quite easy when you plow through them. So wade in and enjoy!
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