I'm trying to fully understand everything instead of just say, "ok, this means that and I'm done"
I'm going to assume you mean you want a breakdown of the grammar mentioned above. So with that in mind:
you like something more than something else
I'm going to take this comment at face value here, so forgive me if I've misinterpreted it.
I would be careful saying AよりBのほう is "liking something more than something else". It can mean that, but that fully depends on the context / sentence intending to mean that.
AよりBのほう is pattern that literally translates to "B's way is more X than A". You right with より meaning "than" and ほう being 方 which means "way" here. The の being used is the の particle and indicates possession. This also means that B would be a noun in this case. Leaving X as being the "more" whatever (e.g. liked, harder, tastier, or etc) indicator.
As for me, I like pizza's way more than hamburgers.
Englishified: I like pizza more than hamburgers.
I guess this also applies to comparing three or more items
Personally, I don't agree with this statement 100% as I don't see の中で as a "hard and fast" intended for comparisons rule. It could be used in this manner assuming the sentence indicates or intends for that.
Xの中で has a literal meaning of "within, among, inside, or etc of X". Working from left to right: X is our thing to consider. The の here is also the の particle indicating possession. 中 (なか) means within, among, inside, or etc depending on the context. And で is the で particle, where it can take on the meaning of with, by, at, via, or etc. This で is also somewhat tricky to translate as it's meaning kind of gets absorbed along with the 中 (なか).
Please wait at the shop's inside.
Englishified: Please wait within the shop.
Notice here we're not comparing anything. The Xの中で is about location.
As for me, with all of the dogs' among, I like Tomato.
Englishified: Amongst all of the dogs, I like Tomato.
However, here Xの中で does have a comparison meaning. We are considering all of the dogs in the scenario and have decided Tomato is the one that is liked