What is the best thing to read if I want to learn set theory? If there is free access to it online, that's a bonus.

I started here ... I don't think I can judge yet whether that's a good script. a few chapters in I took a peek forward and saw that they define the natural numbers as

0 = ∅

1 = {0} = {∅}

2 = {0, 1} = {∅, {∅}}

3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}}

etc.

Is this how proper set theory works – constructing everything through empty sets and sets of empty sets?

Well, there are really two things going on here. First we write down the rules for what arithmetic ought to obey, which we call axioms, e.g. the Peano axioms.

Next, we go about constructing a model of these axioms, that is a set of objects and operations that satisfy the axioms. There are usually lots and lots of different models, and they might not all behave the same way. One model is this one involving nested sets and empty sets.

For another example, in real analysis you usually write down a few axioms for the real numbers, namely those of being a complete ordered field. One model is given by Dedekind cuts, another is given by equivalence classes of Cauchy sequences. Neither model can be said to be the "true" real numbers.

When people say that AC is independent of ZF, they mean that there is both a model of ZF which satisfies AC and a model of ZF where AC fails.