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$newcommand{C}{mathscr{C}}$
Where should one learn about the (?) stable homotopy category? I'll call what we're looking for $C$. There seem to be many competing notions, all of which have some form of equivalence (?). The ones that come to mind are

• 
  $C=hTop$, the homotopy category of $Top$, which can consist of 'nice' spaces such as those that are compactly generated


• 
  $C=hSp$, the homotopy category of spectra (??? see below)


• constructions in EKMM (is this a good place to learn about stable homotopy?)


• Adams' category (said to be outdated)


• Boardman's stable homotopy category (what is this?)


• the stable $infty$-category $C=Stab(S_ast)$, where the right-hand-side is the stabilization of the $infty$-category $S_ast$ of pointed topological spaces. (this is denoted $Sp(S_ast)$ too but I don't want to mix up notation here).

Something that's confusing to me is also the varying notions of spectra. For instance, we have CW-spectra à la Adams, but also sequential spectra $Sp^mathbf{N}$, and the more structured symmetric spectra $Sp^Sigma$. Most of this has to do (I think) with the issue of finding an appropriate smash product $wedge:CtimesCtoC$ which makes $C$ into a symmetric monoidal category.

Can anyone help me understand where I should start?

The homotopy category of spaces is relevant to unstable homotopy theory. All of the other categories are equivalent. If all you want is the stable homotopy category, then Adams' construction is still the most efficient. Note that Adams' construction is just Boardman's, fleshed out. But most people would like a more manageable point-set and/or $infty$-categorical model for spectra, these days.

EKMM had the first excellent point-set construction of spectra, via so-called $S$-modules. However, it seems to me to be a more or less established consensus, even for the authors of that book, that model categories of diagram spectra are a more manageable "in", if you only want one construction. Diagram spectra describe spectra as continuous functors from some topological category into spaces, where the domain can range in complexity from the discrete category of natural numbers (only identity maps) to the entire topologically enriched category of finite CW-complexes. For smaller domain categories, one has to define spectra as functors equipped with an action of a certain ring-functor $mathbb{S}$; when $mathbb{S}$ is a commutative ring-functor one gets the Day convolution symmetric monoidal structure on its modules (very roughly analogous to the tensor product of modules over an ordinary commutative ring) and that gives the nice point-set smash products of spectra to which you allude. As far as where to learn about these things, it's a bit tricky. Adams is still absolutely worth reading. Schwede has an incomplete book project on symmetric spectra that would get you a lot of the basics. Mandell-May-Schwede-Shipley have a nice survey on model categories of diagram spectra to get an overview of what the options are.

While the point-set categories of spectra are very important constructions, one explanation for a relative dearth of thorough expositions on them is that the point-set level details are more important for proving that certain constructions exist than for the majority of actual mathematical work being done in stable homotopy theory. For that, one is often working in the $infty$-category of spectra, which is really canonically defined: for instance, it's the initial stable $infty$-category admitting a homotopy colimit-preserving functor from the $infty$-category of spaces. Every construction of a model category of spectra is a different way of getting at this same $infty$-category. As for references here, well, stable $infty$-categories in general are the subject of Lurie's opus Higher Algebra. However, again, many stable homotopy theorists I know rarely think at this level either. A significant proportion of stable homotopy theory is about trying to compute homotopy and cohomology groups, which eventually often reduces to understanding certain spectral sequences in a very concrete and algebraic way. You could look, for instance, at Ravenel's books to get a bit of an idea of this computational perspective.