GEB is a great book, and I've probably read it at least 3.33333333... times over the years. As a late teen it blew my mind. But I'm not sure I'd recommend it as a route into Gödel's proofs [1]. The book covers a lot of other ground too, and is notoriously digressive and quirky (looking at you, dialogues).

I'd recommend Gödel's Proof by Nagel and Newman for a conceptual intro.

[1] I'm not a mathematician, so my understanding is necessarily informal.

“Godel’s Proof”[1] is also a great and shorter read if the scale of GEB is intimidating(I know it was for me at first).

https://nyupress.org/9780814758014/godels-proof/