In the most reasonable reading, the request being made is impossible. There is no book that:
- Presents an accessible introduction to mathematical logic
- Is reasonably up-to-date and prepares a reader to read other contemporary references in mathematical logic
- Makes no assumptions about the mathematical metatheory, or works with a purely formalized metatheory. Here the metatheory is the set of axioms we use to prove the results of logic that we are interested in.
However, some number of people ask whether there is such a book. So I am going to write this answer to briefly summarize why no book of that form exists.
Why would a book of that kind exist?
The desire for book meeting those 3 bullets tends to be phrased in terms of a circularity between mathematical logic as a subject of study, versus mathematical logic as a tool for studying other mathematical objects. There are several good answers about this circularity at "How to avoid perceived circularity when defining a formal language?".
While there is indeed some circularity, it is not the kind that novices often imagine. Because logicians have had well over 100 years to think about it, the elementary objections that students often pose have all been considered many times. There are several compelling ways to resolve the issue, which is not seen as any kind of major problem in mathematical logic anymore.
Presenting an accessible introduction
It is possible to have a book that pays very close attention to the resources needed in the metatheory. As a concrete example, consider Gödel's incompleteness theorem. Any reasonable introduction to mathematical logic will include some aspects of this theorem.
The book Metamathematics of First-Order Arithmetic by Hajek and Pudlak, and the article "The Incompleteness Theorems" by Smorynski in the Handbook of Mathematical Logic, include the full details about what is needed in the metatheory to prove the incompleteness theorem.
But these two resources are completely inaccessible to someone who does not already know the basics of mathematical logic and the incompleteness theorem. To understand these texts, the readers needs to have a very solid knowledge of informal mathematical logic as seen in introductory texts.
By analogy, it's no different than trying to learn the proof of Fermat's Last Theorem without learning any abstract algebra or number theory first.
At the advanced graduate level, an enormous amount is known about what metatheoretical axioms are required for various theorems in logic (and outside logic). But there is no way to access this knowledge at the introductory level.
Preparing the reader to read more mathematical logic
Imagine a student who wants to learn basic differential geometry, but who thinks that, because of the term "geometry", every argument needs to go back to basic axioms like Euclid's postulates. No book is written like that, if it were even possible - because that is not the way that anyone who actually studies differential geometry would approach the subject.
Similarly, an introduction to mathematical logic is not likely to spend too much time looking at the specific axioms of set theory needed in the metatheory, or at any other formal aspects of the metatheory, because those are niche subjects which many logicians never need to consider at all.
Again by analogy: once we move beyond very introductory levels, books on abstract algebra will freely use terminology and some methods from category theory. A student might ask for a book that does everything in these advanced books without mentioning category theory at all. That could be done, in principle, but a book like that would not prepare a student to read any other literature.
Making no assumptions in the metatheory
There is, actually, a completely formal development of some parts of mathematics: Principia Mathematica by Russell and Whitehead. This book is a stunning landmark in history - and it is completely unread by almost all mathematical logicians today.
On the one hand, a slavishly formal development of any mathematics requires enormous amounts of space. Equally importantly, Principia lacks most of the key theorems that need to be learned in an intro to mathematical logic, not the least of which are the soundness and completeness theorems and Gödel's incompleteness theorem.
Moreover, as I wrote above, a good understanding of the unformalized proofs of these theorems is needed to understand how the formalized proofs work.