The questions assumes that there is some notion of "set" in first-order logic itself, but there is not. We use sets to study first-order logic, particularly the semantics (models) aspect. But these are part of the metatheory we use to study logic, not really part of "first order logic". For example, if we look at the first-order theory of groups, there is nothing in it about "sets".
If we look more at the syntactic (proofs) side, we can get by with a much weaker metatheory, one which only needs to manipulate strings. Theories often used for this purpose include Peano arithmetic and the weaker Primitive Recursive Arithmetic. In these theories, there aren't directly any "sets", just natural numbers, although these theories have ways to talk about functions from numbers to numbers and, as such, indirectly talk about some kinds of sets.
The really fundamental concepts in first-order logic are alphabet, signature, language, theory, formal proofs/derivability, and models/satisfiability. All but the last of these can be very satisfactorily studied using Peano arithmetic as our metatheory. Once we move to studying models - which are again a fundamental part of first-order logic - we usually find it more satisfactory to work in a stronger metatheory that is able to construct and work with models more directly.
On the nature of logic
The other thing about this particular question: it is common for people first studying mathematical logic to think that the main purpose of studying logic is to find the most primitive objects of mathematics and then to rebuild mathematics from these primitive objects -- this is the foundational aspect of logic.
That is indeed one aspect of mathematical logic, but not the only one by far. Historically, the foundational aspect was of particular interest around the turn of the 20th century, but it is not of such primary interest any longer. From the contemporary viewpoint, another purpose of mathematical logic is simply to understand mathematics better by using techniques that have come to be called "mathematical logic". I think that, for historical reasons and because it's interesting, the foundational aspect tends to be slightly over-emphasized in introductory materials.
For example, another common and important thread in mathematical logic is definability - the study of which aspects of mathematical structures can be expressed in which formal languages. This thread runs very heavily through computability theory and model theory, and is also found in set theory and proof theory.
Yet another common thread is an interest in the mathematical objects of logic for their own sake: some logicians study sets because they like sets, not as a way to study foundations. Some study computability because they like computability, without much interest in philosophical aspects. Some research topics in model theory are essentially indistinguishable from abstract algebra or analysis.
The foundational aspect of logic is still important, of course, and there are still people who work primarily on foundations. But the idea that mathematical logic will provide some sort of rock-solid foundation to all the rest of mathematics is not really part of the contemporary study of foundations. Instead we think about a range of theories, each suitable for its own foundational purpose. For studying the semantics of first-order logic, we need a theory that includes some way to handle models, which are particular kinds of sets.
As the shift from a mainly foundational viewpoint to a more broadly mathematical viewpoint occurred, several mathematical logic books from the mid 20th century included detailed explanations in the introduction about why they use advanced mathematical methods to study logic. One good treatment of this topic is in Monk's logic book, which can be found pretty cheaply these days.
The purpose of this section, which may be a slight digression, is to explain that one reason that it is not easy to see how logic is developed "out of nothing" from absolutely first principles is that, often, that isn't the goal that contemporary logicians have in discussing logic. They aren't necessarily trying to develop logic and mathematics from absolutely first principles.
The point is that using ZFC we can interpret1 second-order logic. ZFC gives us a way to understand the notion of sets and relations which are necessary for second-order logic.
We can prove, as a first-order statement in the language of ZFC, that there is a unique Dedekind-complete ordered field (up to isomorphism). This is a second-order statement about the real numbers, yes, but we can interpret that in ZFC, which transforms it into a first-order logic statement in the language of set theory.
1. In the non-technical sense.
Best Answer
You could start from this Study Guide to logic books as a pointer to some of the best that are available, and then you can pick what suits you.
Chapter 3 is the key one for you. The warmly recommended Friendly Introduction ... is now freely available as a PDF download, courtesy of the authors.