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).

Instead 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/

It’s actually an interesting fact that every person who was programming in the 80s owns a copy of GEB, which they put on the bookshelf and never actually read.

while it is a groovy into to recursion and other cool ideas, GEB annoys me in that I feel like the three figures in the title are ill matched. Godel proves a super important result in math, sure... Escher was a skilled draughtsman who had a feel for tesselation. An OK artist IMO but no special insights. Bach on the other hand was an expressive genius who in the volume, power and beauty of his productions just seemed to drop out the sky like a meteor. Escher does not belong in the same breath frankly. if Bach made a crab canon or did other marginally math-y things that is just not the point - the work lives or dies in entirely different terms...