1. I don't think there is any reason to suppose that learning about mathematical logic from Principia to Gödel will be any help at all in understanding what is going on with the Frankfurt School. (The only tenuous connection I can think of is that the logical positivists were influenced by developments in logic, and the Frankfurt School were concerned inter alia to give a critique of positivism. But since neither the authors of Principia nor Gödel were positivists, it would be better to read some of the positivists themselves if you want to know what the Frankfurt School were reacting against).
2. Of course, I think finding out a bit about mathematical logic is fun for its own sake: but it is mathematics and to really understand I'm afraid there is not much for it other than working through some increasingly tough books called the likes of "An Introduction to Logic" followed by "Intermediate Logic" and then "Mathematical Logic". Still, you can get a distant impression of what's going on by following links on Wikipedia etc. And on Gödelian matters, Hofstadter's long book is entertainingly illuminating and somewhat annoying in about equal measure. Goldstein's book, though, is hopeless as a guide: see http://math.stanford.edu/~feferman/papers/lrb.pdf for an authoritative demolition (which indeed pulls its punches). If I was going to recommend one book on Gödel as a way in for the non-mathematical, it would be Torkel Franzen's Gödel's Theorem, an Incomplete Guide to its Use and Abuse.