I've always found that learning new areas always goes a lot better if you start with a key insight of what the field is about. Often this is not presented or explained at the beginning of the course, and you have to deduce it later on.

For instance, I would have better grasped the epsilon-delta definition of a limit if the instructor had started with something like:

• Our intuitive definition of a limit is that as we get closer to this point, the function gets closer to this value. It has turned out to be very tricky to formalise this intuition, however. Early mathematicians used calculus without a good definition of limit, and their informal definitions led to a lot of paradoxes. The epsilon-delta definition is a bit clunky and may seem counter-intuitive, but it actually manages to capture our intuitive definition without paradoxes and problems - that's why we choose it, not for its elegance (though you will come to appreciate it). With that in mind, let's have a look at it...

Similarly, I would have made more rapid progress with Gödel's theorems if, before giving the formal definition of Gödel numbering and of the provability symbol □, someone had clarified that direct and indirect self-reference was a problem. If a formal system of a certain complexity can talk about its own structure, even without "realising" that it's doing so, problems will arise. Some of my other key insights in the field can be found in my post here.