On "familiarity" (or How to avoid "going down the Math Rabbit Hole"?)

Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole." For example, suppose you come across the novel term vector space, and want to learn more about it. You look up various definitions, and they all refer to something called a field. So now you're off to learn what a field is, but it's the same story all over again: all the definitions you find refer to something called a group. Off to learn about what a group is. Ad infinitum. That's what I'm calling here "to go down the Math Rabbit Hole." Upon first encountering the situation described above one may think: "well, if that's what it takes to learn about vector spaces, then I'll have to toughen up, and do it." I picked this particular example, however, because I'm sure that the course of action it envisions is one that is not just arduous: it is in fact utterly misguided. I can say so with some confidence, for this particular case, thanks to some serendipitous personal experience. It turns out that, luckily for me, some kind calculus professor in college gave me the tip to take a course in linear algebra (something that I would have never thought of on my own), and therefore I had the luxury of learning about vector spaces without having to venture into the dreaded MRH. I did well in this class, and got a good intuitive grasp of vector spaces, but even after I had studied for my final exams (let alone the first day of class), I couldn't have said what a field was. Therefore, from my experience, and that of pretty much all my fellow students in that class, I know that one does not need to know a whole lot about fields to get the hang of vector spaces. All one needs is a familiarity with some field (say $\mathbb{R}$ ). Now, it's hard to pin down more precisely what this familiarity amounts to. The only thing that I can say about it is that it is a state somewhere between, and quite distinct from, (a) the state right after reading and understanding the definition