The more advanced topics have been discussed in existing answers, so I'll talk about introductory/general topology instead.

I actually disagree with Ron Maimon's assessment of point-set topology in the comments -- the "analytic" language of point-set topology is natural, it comes from the fact that topology naturally arises as a generalisation the analytic idea of a limit. Specifically, the limit is the "underyling structure" of a topological space -- a map is a homomorphism (note the spelling) of topological spaces, i.e. a continuous map, if it preserves all limits. So for physicists at least, a continuous map is just one that commutes with the limit operation:

$$f\left(\lim_{x\to a}x\right)=\lim_{x\to a}f(x)$$

This is a perfectly sensible "structure" to talk about. It's also perfectly sensible to stop here, but mathematicians like to abstract and generalise things -- the natural way to generalise limits to general spaces is from the notion of a "filter", due to Bourbaki, which leads to the axiomatisation of topology in terms of neighbourhoods. The open set axiomatisation then arises as something to prove to be equivalent to the stuff with neighbourhoods (and the intuition for why this makes sense is that openness and closedness are fundamentally connected to the idea of a limit in a more general setting than sequences -- perhaps a good intermediate axiomatisation to understand this is Kurwatzki's closure, which is precisely the notion of such a generalised limit operator).

As for resources, I wouldn't be too focused on trying to find the best book on the subject, because (1) it's really your introspection about the subject that helps you learn stuff (2) literally everyone in the world has worked with topology and you shouldn't find it too hard to find some content to learn from.

But there are some important insights you should make sure you acquire, and the series of articles I'm writing here are optimal for that: https://thewindingnumber.blogspot.com/p/topology.html. Not everything is done yet (particularly the algebraic topology stuff), but it's at least somewhere to start.