I also took a Computability course. We used:

https://www.amazon.com/Computability-Logic-George-S-Boolos/d...

I can't speak for Godel's original proof since I haven't seen it before, but I certainly found Computability and Logic [1] to be pretty approachable. It's the textbook used for UC Berkeley's "Intermediate Logic" philosophy course (in other words, it's so easy even philosophy majors can understand it! :P).

From my understanding, C&L diverges from Godel's original proof technique in order to make it easier to follow, but it's much more rigorous and explicit than what you'd find in Godel, Escher, Bach or something. It's still a textbook.

[1]: https://www.amazon.com/Computability-Logic-George-S-Boolos/d...

I have been through a slew of logic texts. By far, my favorite for both theory and practice is Computability and Logic by Boolos, Burgess and Jeffrey.

http://www.amazon.com/Computability-Logic-George-S-Boolos/dp...

Another very good book on nuts and bolts proofs in both propositional and predicate logic is Beginning Logic by E. J. Lemmon.

http://www.amazon.com/Beginning-Logic-E-J-Lemmon/dp/09151445...

And a great collection of primary texts covering the history of modern logic is From Frege to Godel..., edited by Jean van Heijenoort.

http://www.amazon.com/Frege-Godel-Mathematical-1879-1931-Sci...