In favor of the rabbit hole
Begin at the beginning, and go on till you come to the end: then stop.
My answer is somewhat contrarian, but I do believe it strongly: You have to fall into the rabbit hole, and you have to go as deep as possible at all times.
Math isn't science. To understand why a frog croaks we might study its respiratory system, to understand this we might get into how cells divide, then the chemical processes in living organisms, organic molecules in general, then the physics of atoms, the sub-atomic particles, etc. By now we have certainly gone too far. The best way of understanding how a frog breathes is to make simplifying assumptions at much higher scales. So we assume that cells are little machines that do a certain thing, or at least that atoms are small balls that bounce off each other and stick together.
To understand vector spaces, there is no point trying to 'gloss over' fields. Best is to know fields inside out. If you can't learn them inside out, you should know the basics as well as possible. The rabbit hole doesn't go on forever - in all cases you will reach either basic definitions, or things that we all agree on intuitively (like the counting numbers).
Sentence first, verdict afterwards
I have adopted a discipline of studying mathematics, where I never go past a single word that I don't know what it means, and I never skip over a statement that I can't understand, or justify, or understand the justification of. As soon as I have a question of the type "but why wouldn't that work if that condition wasn't true", I stop and think about it until I understand it. If I need to go back earlier in the book or to another book, I do so, even if it means I end up reading books backwards.
I don't always work like this, and there could be places where it is unnecessary, but your example definitely isn't one of them. Suppose that you are trying to learn about vector spaces. Frankly you aren't doing yourself any favors if you believe deep down that all fields are $\mathbb{R}$ and all vector spaces are $\mathbb{R}^n$ for some small natural $n$. Each time you justify something or try to picture something using this model you are storing up problems for yourself when you encounter infinite dimensional spaces, finite fields, or spaces over $\mathbb{C}$ and your mental models don't work anymore. Not to mention the huge problems you would have encountered if you had some misconception about a field (a field is a special case of a group, or a field is an ordered set with some other properties...) and had continued studying vector spaces for a while with this wrong picture in your mind.
Obviously you wouldn't adopt this method to read a newspaper, or a history book. If you don't know what an arquebuse is, but you have an idea that it's some kind of weapon, you might as well assume it's a type of sword. When you find out it's a kind of gun, you can simply slot this knowledge into the understanding of whatever you were reading about arquebuses. Similarly if you think Wisconsin is in Canada. Either it doesn't matter to the story which country it is in, or it does matter, and you find out soon enough where it is, with little effort wasted in reinterpreting other parts of the story.
Curiouser and curiouser
Now suppose that you do follow my method. You start to research fields. The information about fields you need is this:
- You need to know the axioms that define a field. These easily fit on a sheet of notepaper in large caps written in sharpie. All of them are pretty much self-explanatory. Knowing the names (commutativity etc.) will be invaluable. Arguable any pure mathematician should know them.
- You should know several examples of fields, eg $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{F_p}$. If the p-adics or other more advanced fields are accessible to you, then all the better, but if not you can live without them. Again, there is nothing here that is not worth knowing to anyone in any branch of pure maths (or in other words, this material is something all undergraduate mathematicians have to learn before they specialize).
- You should know some examples of things that are a bit like fields, but not fields. Eg $\mathbb{N}$, $\mathbb{Z}$, residue sets modulo a composite, reduced residue sets, $\mathbb{R}^n$, boolean rings. There is nothing extra to learn here, you only need to be able to check that various objects you have already heard of don't satisfy the definition of a field. If you haven't heard of some space, you don't need to know that it's not a field. In each case you just figure out one line of the definition that you can point to and say 'This fails, therefore not a field'.
- You should be aware of some theorems about fields. In particular, you should know which properties of 'numbers' apply to all fields. This is again just a case of chasing the definition of a field through some knowledge you already have. Is the quadratic formula applicable in any field? No because square roots are not part of the definition. Is the formula for the solution of a linear equation applicable in any field? Yes, because the multiplicative inverse must exist because...
- You should be aware of some differences between fields. Which fields are finite? Which fields are complete (if you already know what complete means)? What is the characteristic of a field (a one line definition)?
- You should understand how a field has two groups embedded in it.
Now, in all of this there are really only three things which could cause you to get pulled further down the rabbit hole. $\mathbb{C}$, $\mathbb{F}_p$, and the definition of a group. All of these are things that everyone should know. In the case of the first two, all you need to know is the how the field structure works, not any other properties. This can be worked out or taught in an hour (how to add, how to multiply, how to divide, in $\mathbb{C}$ or $\mathbb{F}_p$). As for groups, you need to know the equivalent of the list above for fields. But you would honestly not be wasting your time if you were to read a whole introductory book on groups, studying every proof in detail, even if your goal is vector spaces. A vector space is also a group, as are half of the next thousand spaces/objects you are going learn about. Furthermore, as you learn about groups you are learning:
- how abstract algebra works: you don't assume inverses, commutativity, etc unless the axioms say you can.
- how to do logical proofs, eg how to prove that two sets are equal
- notation in abstract algebra, eg writing binary operations as sums, as products, or in some other way.
- some examples of groups that you can later think about when you are trying to consider vector spaces or fields in generality (for instance, why can you add a multiplicative structure to some additive groups but not others).
All of these are going to help you massively. Even if you NEVER get to vector spaces (you will), it will have been worth you while. Someone who understands groups is a better mathematician than someone who thinks they know what a vector space is, but doesn't understand groups.
Now, instead of thinking of $\mathbb{R}$ everytime you read a fact about a vector space, you can think of vector spaces over all the fields you know. Rather than limiting your imagination, you are challenging it to understand the statements you read in as close to full generality as possible.
Did I load my example? If you are studying something more specialized, then the 'rabbit hole' subjects are not things everyone needs to know, but they are things you need to know. If you are studying Brownian motion, any fact about elementary probability you come across is something you should be able to master.
Everything's got a moral, if only you can find it
When we speak to professors and other experienced mathematicians about difficult mathematical concepts, they are often able to summarize them with beautiful and compelling generalizations ('Harmonic analysis is about dualities between smooth behavior at small scales and convergent behavior near infinity'). This often makes us worry, because we aren't able to figure these things out ourselves, and they don't appear in our elementary textbooks.
The person who can make that generalization isn't fluent at doing manipulations and solving problems about Fourier transforms because she has understood 'what harmonic analysis is about'. She is fluent, AND she can come out with nice generalizations, because of all the work she has done on the mechanics of harmonic analysis, solving problems, reading slowly through proofs etc. To emulate this person, we shouldn't try to be able to summarize a topic in a neat way. We should try and acquire their detailed knowledge of the mechanics of their subject. Once we've done this, the flashes of insight will come by themselves.
You should also know that some harmonic analysts might think that this 'motivation' for harmonic analysis is wrong, irrelevant or trivial. However all kinds of harmonic analysts would be able to follow each others' work by giving the precise definitions they are working with, starting with the things they all agree with.
Seeing the big picture is nice, but you can often afford to miss it. Miss the small picture, and you're not doing mathematics anymore, just reading about it.