@PauloHenriqueL.Amorim Rautenberg is indeed a good book, but it is not suitable for beginners. In my opinion, if you truly want to understand logic, you must first learn to use a formal deductive system for FOL to 100% familiarity, such as the Fitch-style natural deduction variant linked from my profile:

The reason is simple. One cannot study logic without using logic, and if one is unfamiliar with formal logic then it is almost impossible to be clear about what is going on. Indeed, before FOL you of course need to learn PL (propositional logic), and after FOL you still need to learn to use it in actual mathematics, for which I always suggest PA (Peano Arithmetic). It will only take a few weeks for you to learn all that, if you set your mind to it!

I want to emphasize that those who cannot easily construct formal proofs over PA cannot have any real understanding of Godel's incompleteness theorems, and there is really no shortcut. There are many popular books out there that claim to tell you about Godel's incompleteness theorem but most of them are either wrong or so misleading that it is better not to read them. "Godel without (too many) tears" by Peter Smith is a very good book, but assumes full familiarity with basic FOL.

@PauloHenriqueL.Amorim: If you would like to start learning FOL, the best way to do so is to get your hands working on some exercises. I don't really believe in just reading books; one cannot learn to be an engineer by reading lots of engineering books. Feel free to try out the exercises I have just posted here, and post your attempts in that chat-room.

Thanks for pointing this exercises and the invite for posting attempts here, For the moment I keep going through the implicational calculus exercises in the book I told before. for instance today I was going through basic theorems like $(P) \Rightarrow (P)$ ... But using Kleene's Axioms, as the author told Kleene's, Tarski and Lukasiewicz axioms for implicational calculus lead to the same system because Axioms in someone are Theorems in others.

I did not read the post your linked yet, but its clearly by the organization its very complete, so thanks an advance for sharing this knowledge... You seems to be very experienced and probably know better than me how many times its hard to find good resource in a clear and organized way in the same place.