The best text to study both incompleteness theorems

Speaking as a beginner myself:

I haven’t read it all yet, but An Introduction to Gödel’s Theorems by Peter Smith seems like a good candidate, and it doesn’t have many prerequisites. Smith also wrote a series of shorter handouts on the topic, Gödel Without (Too Many) Tears.

There’s also Godel’s Theorem: An Incomplete Guide to Its Use and Abuse by Torkel Franzén, which is much less technical and primarily concerns false myths about the incompleteness theorems; in my opinion, it is a good companion (not a substitute) for Smith’s book.

First chapter of Jean-Yves Girard, "Proof Theory and Logical Complexity", Vol I, Bibliopolis, 1987

It satisfies all of your conditions, but it is not an elementary book. If I remember correctly, the authors (A.S. Troelstra and H. Schwichtenberg) of the book "Basic Proof Theory" which is published in 2001 wrote in their introduction that their intention was to fill the gap between this and all other (introductionary) books in proof theory. As far as I know, he never published the second volume.

For instance, there is a well-regarded recent book of Torkel Franzen:

Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse

A detailed and positive review was given by Panu Raatikainen in the Notices of the AMS.

Honestly, your question seems underdetermined, since there are many other well-regarded books that an internet search will reveal to you. I would suggest just picking one and trying it out.