What is the best/most effective way to gain experince in using set theory to understand math, calculus in particular? Note: While this question as phrased in the title is somewhat subjective, what I'm looking for as an answer should be specific enough to still have a clear/valid answer(s) to it. Also, apologies in advance if this is somewhat long winded but I feel the full context of my question is needed for clarity. 

First let me provide some context. When it comes to learning new areas of math (or in this case, arguably, more fundamental/abstract areas), one problem all students face is 'vision'. Taking new concepts and adding them to old ones is often difficult since to do that requires seeing how things interrelate. However your typical education in math usually leaves you vastly unprepared to do this.

In a sense, most modern math education teaches students backwards. Most of the maths that students learn is gutted of the more important (and abstract) details. Details such as the 'real' proofs of certain theorems being replaced with a shorthand version that, while proves the part of the concept being focused on, ignores the parts that give it its foundation. "Proofs in a vacuum" as I call them since they usually don't bother proving (sometimes not even mentioning) other concepts that the main idea depends on. 

This leads me to my problem. I have a rather strong foundation in multiple areas of math (particularly those often relevant to physics, such as calculus), but since my education has focused on results/applications rather then math as a whole I'm left without the 'glue' that ties it all together. 

The language of set theory (as well as basic forms of logic) is still new to me. While I have a strong (enough) foundation in the basics after recently completing a class in discrete math, taking the concepts (and building onto them where needed) and applying them so as to add to my understanding of calculus for example is still difficult. 

My question is this: What are some methods/pet projects I should try an implement to 'break out' of this novice phase as far set theory is concerned? To bridge the gap between knowing basic ideas concerning sets and relations between them and concepts from calculus for example? I don't want to look at an equation and see an equation. I wanna be able to see it as a more abstract entity. 

Keep in mind what I'm looking for in an answer is:


Specific topics to research that show how sets and relations are used to construct other others of math
Go to examples for me to play around with so as to see how set theory is applied in practice (for example, how relations can be used to construct basic operations like addition, or differential operators).
Examples of a common (but not obvious) mistakes people tend to make applying set theory to describe common concepts in other area of math.
If I understand your question correctly, although you have embedded a critique of how abstract math is taught, your question is really about how you can shore up your own understanding of abstract math via an enhanced understanding of set theory.

I would suggest that you learn about how the real number system is built up in steps from Peano's axioms for the natural numbers. There are three steps:


From Peano's axioms for the natural numbers to the integers.
From the integers to the rational numbers.
From the rational numbers to the real numbers.


There is one still deeper step:


From the ZFC axioms of set theory to Peano's axioms for the natural numbers (via the construction of the ordinal numbers).


I cannot recommend a single source to you for all four of these steps, although steps 2, 3, 4 at least are in many books at the advanced undergraduate or early graduate level, with steps 2 and 3 more likely to be in number theory books and steps 3 and 4 more likely to be in advanced calculus books. For Step 2, I like the closing two chapters of the book "Theory of Numbers" by B. M. Stewart. For Step 1, I like the early sections of Cohen's slim little atom bomb of a book "Set theory and the continuum hypothesis".
Ultimately, I believe you will find your task to be difficult.

The problem lies with Zermelo's emphasis of Fregean principles over Cantorian principles and the subsequent criticism of Zermelo by Skolem.  

Applied mathematics relies fundamentally on Liebniz' principle of identity of indiscernibles as expressed within the axioms for metric spaces.  Logicians and analytical philosophers have rejected this as a logical principle despite the fact that this rejection leads to contradiction when the sign of equality is interpreted as mere substitutivity.  For them, the priniciple of identity of indiscernibles constitutes an epistemic warrant for uses of the sign of equality.  But, they are concerned with extensional semantics.  So, in first-order logic, the principle of identity of indiscernibles is taken to constitute "real definition".  This is because the logic carries what is called "existential import".  The objection is intended to admit the possibility of two distinct objects having all of the same properties.  Thus, 
they cannot be discerned from one another by their properties.  The claim is that a definition cannot determine an individual.

If you look carefully at the axiom of extension in standard texts on set theory, you will find that it uses a conditional rather than a biconditional.  The conditional defers to the treatment of identity in first-order logic.  And, the identity of indiscernibles is not part of the 
standard account of identity for that logic.  One simply has that $x = x$ means "x is substitutable for x" and that $x = y$ means "y is substitutable for x".  What an applied mathematician usually understands as an equivalence 
relation can be derived from these rules with substitutivity tricks.

This interpretation for identity in set theory follows from Skolem's criticism.  In Zermelo's original axioms, the sign of equality is interpreted with respect to the singleton sets in the theory.  As for Zermelo emphasizing Fregean views over Cantorian views, there is a difference between "the extension of a concept" and "a collection taken as an object".  There is much in the historical development of set theory that has given it the form that it has.  The fact that Cantor's views had been set aside is one of the points which Lawvere has emphasized in his development of sets using category theory.  In his work, the identity of indiscernibles is taken for granted as a necessary logical law.  And, given that Cantor's notion of set had been closely bound to his topological ideas, Lawvere's program should not be dismissed merely because analytical philosophers have different priorities from a significant number -- maybe even a majority -- of mathematicians.

I strongly suggest that you look at Max Black's discussion of the principle of identity of indiscernibles in the paper,

http://home.sandiego.edu/~baber/analytic/blacksballs.pdf

It may help you to decide which conception of set will be more amenable with your objectives.

And, there is one other point to make.  You speak of sets in the constructive manner with which they had been introduced historically.  The notion of formalism in mathematics has deprecated this relationship to other parts of mathematics.  This is largely why Skolem's criticisms had been accepted.  One may speak of how mathematical objects may be represented within set theory, but, set theory is not as widely considered to be "foundational" as it had been in the past.  This is largely because of how arithmetic may be used to represent statements about what can and what cannot be proven.  But, that is different from what is meant by formalism, and, formalism divorces set theory from the role of being "glue".

Modern set theory is simply not what most applied mathematicians think it is.  One may "believe" in sets, but the contentiousness of the positions in Professor Black's paper may very well have divorced set theory from applied 
mathematics.  And, this insistence on a logical aesthetic at the expense of mathematical forms may be traced back to passages from Aristotle.

What you seem to be looking for is a single progressive development leading to elements used by applied mathematics.  I have found "Vectors and Tensors" by Bowen and Wang,

https://www.amazon.com/Introduction-Vectors-Tensors-Two-Mathematics/dp/048646914X

to be useful in this regard.  There is an online version if you look for it.  It comes in two volumes, and, the transition from algebraic development to metric spaces is in the first chapter of the second volume.  It is this transition which should motivate the study of Willard's self-verifying systems of arithmetic rather than Peano arithmetic.  But, unless one is particularly interested in how foundational studies relate to applied mathematics, one would not even recognize this. 

While I hope to have given you alternative ways to think about your situation, I suspect that you will still pursue set theory out of your own curiosity.  Good luck.