I started math four months ago with modules like measure theory and topology. It was unavoidable to notice how many concepts are more general than what I thought before.
For example the $\varepsilon,\delta$ definition of continuity in $\mathbb{R}$ is less general that the $\varepsilon,\delta$ definition on a metric space, and even more general is the preimage definition in a topological space. In this case it seems to me that the last definition is the one that really contains the true concept, the ESSENCE of continuity.
This made me think that, notwithstanding the fact that (unavoidably) these concepts are taught more or less the via their historical development, with all its advantages, I should try to get into the mindset that, following the previous example, cointinuity is a matter of preimages of open set, and it follows that this is the definition for checking the continuity of $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x$, say, and it luckily happens that it can be equivalently written as the $\varepsilon,\delta$ definition (considering the luck of the equivalence going the other way around).
I.e. I should try think about mathematics as if I started learning it from the most general concepts instead from peculiar examples as it happens in high school.
Question: is this mindset the right one I should try to get into?
The answer is yes and no. On the one hand, you should certainly try to understand the general definition in order to make progress. On the other hand, you should keep in mind that a typical property or concept in mathematics can be generalized in various ways. What seems like the most generous definition may end up being less fruitful because it does not in the end express the essense of the original intuitive concept. For example, the "right" definition of compactness is generally thought to be the one in terms of choosing a finite subcover of an arbitrary finite cover. But if you try to prove theorems with this definition in mind you might find this unsetting. The initial intuitive insights should always be retained while one is making progress. Another example: if you try to do analysis while thinking that a real number is an equivalence class of Cauchy sequences of rational numbers, you might quickly need to take an aspirin or worse :-)