TITLE: On "familiarity" (or How to avoid "going down the Math Rabbit Hole"?)
QUESTION [426 upvotes]: 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 of whatever it is one wants to learn about (say, "vector spaces"), and (b) the state right after acing a graduate-level pure math course in that topic.
Even harder than defining this familiarity is coming up with an efficient way to attain it...
I'd like to ask all the math autodidacts reading this: how do you avoid falling into the Math Rabbit Hole?  And more specifically, how do you efficiently attain enough familiarity with pre-requisite concepts to move on to the topics that you want to learn about?
PS: John von Neumann allegedly once said "Young man, in mathematics you don't understand things. You just get used to them."  I think that this "getting used to things" is much of what I'm calling familiarity above. The problem of learning mathematics efficiently then becomes the problem of "getting used to things" quickly.
EDIT: Several answers and comments have suggested to use textbooks rather than, say, Wikipedia, to learn math. But textbooks usually have the same problem. There are exceptions, such as Gilbert Strang's books, which generally avoid technicalities and instead focus on the big picture. They are indeed ideal introductions to a subject, but they are exceedingly rare.  For example, as I already mentioned in one comment, I've been looking for an intro book on homotopy theory that focuses on the big picture, to no avail; all the books I've found bristle with technicalities from the get go: Hausdorff this, locally compact that, yadda yadda...
I'm sure that when one mathematician asks another for an introduction to some branch of math, the latter does not start spewing all these formal technicalities, but instead gives a big-picture account, based on simple examples. I wish authors of mathematics books sometimes wrote books in such an informal vein. Note that I'm not talking here about books written for math-phobes (in fact I detest it when a math book adopts a condescending "for-dummies", "let's-not-fry-our-little-brains-now" tone). Informal does not mean "dumbed down".  There's a huge gap in the mathematics literature (at least in English), and I can't figure out why.
(BTW, I'm glad that MJD brought up Strang's Linear Algebra book, because it's a concrete example that shows it's not impossible to write a successful math textbook that stays on the big picture, and doesn't fuss over technicalities.  It goes without saying that I'm not advocating that all math books be written this way.  Attention to such technical details, precision, and rigor are all essential to doing mathematics, but they can easily overwhelm an introductory exposition.)

REPLY [2 votes]: I think Halmos's opinion is extremely helpful in these situations: "When you come to an obstacle, a mysterious passage, an unsolvable problem, just skip it. Jump ahead, try the next problem, turn the page, go to the next chapter, or even abandon the book and start another one. Books may be linearly ordered, but our minds are not."
If you follow this advice, you would later find that the obstacle is either not important to the subject (i.e. you recognizing that the definition of a "field" is not important to linear algebra) or you would have more information on the challenge, so it's generally easier when you revisit that part of the book for review purposes. This would save you a lot of time focusing on unimportant details and is better for a holistic understanding of the subject.
Finally, if you found yourself skipping a little too much, it's probably worthwhile to think about whether the book (or the subject) is suitable for you. Ask a professor that knows your background or have a look at stackexchange for book reviews. With that said, I don't think linear algebra has this problem and all you need is a good book that caters to your way of learning.