I want to know if anyone would recommend some website/book/source where I can begin learning about topology. Here is my situation:
Currently the highest level of math which I can take is a class on mathematical analysis/real analysis, which only briefly covers topology.
I want to know what sources I should use to introduce myself to topology and then go as far as I choose, and not be limited by material I have available.
Take analysis to get a sense of limits and continuity. This will give intuition for topology which is a generalized notion of limits and continuity. Then study point set topology as suggested by GNU. Point set topology is about the texture of a space. Afterwards you could study algebraic topology which is about shape of flexible spaces such as does it have holes or knots. You will want a basic understanding of abstract algebra, at least groups, for algebraic topology.