I started off my (never finished, might restart it here on TS) Poincaré Project with the following:
Thinking Like a Pure Mathematician
Before we are at a point where we can discuss the Poincare Conjecture itself, we need to learn some general topology and group theory. But before we lay that foundation, I think it is worth taking a moment to establish the mode of thinking we must enter.
Marcus Aurelius exhorts us to ask "what is the nature of the whole, and what is my nature, and how this is related to that, and what kind of a part it is of what kind of a whole?" Now Aurelius is talking about human nature (and see Hannibal Lecter's use of the quotation in Silence of the Lambs) but it encapsulates the fundamental questions asked by pure mathematicians, not of humans, but of abstract objects such as numbers and shapes.
Imagine you're looking at an apple and you notice certain characteristics it posseses. Which of those characteristics are specific to that particular apple? Which are specific to all apples of that particular variety? Of apples in general? Or of all fruit? Of food? Organic objects? Physical objects?
In mathematics in general, and in the early days of this Poincare Project in particular, we will often be asking questions like: what is the most general object that exhibits this characteristic? What is the distinguishing characteristic of this object compared with others we're dealing with?
Get your mind in a mode to ask those kind of questions and we'll be ready to introduce topology.