First of all, Gödel's theorems are not a part of set theory, they are rather a part of mathematical logic.
If you just finished book of proof, then you are likely not ready to tackle Gödel's theorems or axiomatic set theory. I'm sure you could formally understand the notions though. But you wouldn't understand the underlying motivations and intuition. For example, why was axiomatic set theory necessary, and what exactly is Gödel's theorems about? This requires some familiarity with the rest of mathematics. Furthermore, if you finished the book of proof, then you are likely not very skilled in proofs yet.
So what I propose is to first learn a bit on abstract algebra and real analysis. This will help you greatly to obtain more intuition for set theory and Gödel. Furthermore, I encourage you to read up on the history and philosophy of mathematics. Gödel is merely the last chapter in a story that has been going on since the ancient Greeks! So knowing the full story is important.
Some good books to read:
Pinter's "A book of abstract algebra"
https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20
This is necessary because abstract structures such as groups and rings will be generalized in mathematical logic. So it is good to be acquainted with such structures first, before seeing the generalization.
Bloch's "The real numbers and real analysis"
https://www.amazon.com/dp/0387721762/?tag=pfamazon01-20
This book is good because it develops mathematics from the ground up. It only accepts sets as given, and then it builds the natural, integers, rational numbers and real numbers. It goes on to develop most of calculus rigorously. This book is necessary because you need to understand what questions set theory wanted to answer. Knowing the Peano axioms and the construction of the real numbers is absolutely crucial for that. Not an easy book though!
Stillwell's "The real numbers : an introduction to set theory and analysis"
https://www.amazon.com/dp/3319015761/?tag=pfamazon01-20
This book starts with some very foundational questions important to set theory, and then answers them. Stillwell is a great writer too!
Kline's "Mathematical Thought from Ancient to Modern Times, Vol 1 and 2"
https://www.amazon.com/dp/0195061357/?tag=pfamazon01-20
Very comprehensive history of mathematics. Very worth reading. You don't need to read all of them of course, but it's important to get a historical grasp on what questions were important in set theory.
Gensler: "Gödel's theorem simplified"
https://www.amazon.com/dp/081913869X/?tag=pfamazon01-20
If you're interested in Gödel's theorem (or one of them), then there is no better book to start than this. It is quite easy, but very worth reading.
Do not hesitate to contact me for more questions or for guidance!
