How to develop intuition in topology?
In my opinion, one of the best ways of developing intuition in topology is to study other branches of mathematics in which there are topological spaces. Many of these definitions, properties and theorems were imagined by people who were working in related branches of math, mostly analysis and geometry. These people stumbled upon spaces which had remarkable or singular properties, so they studied these properties. The examples came first.
Why does anyone care about compactness, say? The best way to answer this question is to ask the question: who were the first people to care about compactness, and why did they? What kind of spaces were they working with?
If you want to learn a new language, there is no point in reading the dictionary and the thesaurus. There is also not much point in learning all of the bizarre exceptions before you encounter them naturally. Instead, you should learn a few basic principles, and then go out and talk to people. Figure out how they speak, and refer to the thesaurus as you go along.
Read "Explorations in Topology" by David Gay. I'm too in search for understanding in topology and this was recommended by my adviser (topologist). So far the book is more than expected. It's amazing.
Though I don't have any authority on this specific subject matter, I would say the thought of looking for a "trick" or "shortcut" for anything is quite dangerous. Just put the head down and work laboriously is the best approach to everything. "A couple of months" is obviously extremely short time. To really gain any basic understanding about anything you would take at least 3 years or even more. To master one? Probably 10 years or more. In everything I learned, "exercises" are probably exactly the thing that helped me gain "intuition", if any. If you always think for "shortcut" or "trick" you can very well be unintentionally slowing yourself down by distracting yourself too much. This is just a reminder; hopefully I won't be bashed too much for this unanswerlike answer :P
I also don't agree with Joyal's perspective on learning languages. Of course you go and talk to people. But in fact mastering the usage of words and special instances through enough exercises alone will immensely speed up the process. That's "achieve multiple times of effect in half the time" in Chinese. If you don't have a decent command of quite a few words and phrases you're never gonna understand anything no matter how much you "talk". That's why paying unbalanced attention at any of those both aspects will lead to disasters.