Value in retracing mathematicians' steps (specifically Galois)? So I'd like to learn Galois Theory, which I am probably not "qualified" for in an ordinary sense (I've never done abstract algebra, and I'm just now learning linear algebra in my vector calculus course), but the way I'd like to approach it is to try to sort of retrace Galois' steps, which, my thinking is, necessarily implies that I wouldn't need abstract algebra, because Galois certainly didn't have Dummit and Foote open for reference while he was messing around with permutation groups.
There are a couple of reasons why I'd like to do this, and a couple of reasons why it might be a bad idea, so I'm just trying to get the opinions of people who actually know what Galois theory entails. Here are the reasons why I want to try this:
1) I feel a gap in my education where high school "algebra" is concerned. You don't (or I should say, "I didn't") really get introduced to the idea of proofs and the abstract "why it works" of mathematics until calculus, so most of my algebra education consists of "here's how you can factor this particular quadratic equation," or "here's this neat trick for xyz." And this is all stuff that Galois, Abel, Gauss, etc., were wrestling with on the advanced, abstract level that I'd like to acquaint myself with.
1a) I expect that I will really like abstract algebra/group theory and that algebra, in some form, could very well be where I end up mathematically, so it seems like this is one of the most important areas to clear up and really understand on a deeper level.
2) (This is not all that distinct from 1 or 1a.) I get the impression that the "discovering" part of mathematics is essential to the abstract viewpoint. That is, in much the same way the Wittgenstein wrote that he doubts people will understand the Tractatus unless they've thought the stuff already, I get the feeling that abstract mathematics is best understood after you've come up with the raw thoughts on your own. So the idea would be that, given that group theory at least partially had its start in Galois' ideas, I feel like this would be a good way to get that background for group theory in the general sense.
3) This is less relevant for the actual question so feel free to ignore it in your response: I'm a little tired of just being walked through mathematics (when I say "walked through" I picture a dog on a leash being walked through a maze without being given a chance to explore and find the dead ends and whatnot). I like what I'm learning, but I feel like neither in college (currently a freshman) nor high school are we given the time (or are expected) to stop and follow a line of thought or just really understand the concepts we are learning. I realize that math classes are structured like this because it's probably a more efficient way to accumulate a wide array of important tools for later. But I don't want to miss the part that's drawing me to math: the understanding part, the finding-the-structures part.
(So (3), is probably outside of the realm of this question but I just thought I'd give some context.)
For (1-2), my main question is "where to start?": 
I was thinking of reading some of Lagrange's stuff ("Résolution algébrique des équations" and maybe some other stuff from his oeuvres... It's all - incredibly - online... seriously astounding that you can find all of it online...) and maybe something from Abel's oeuvres (as per Spivak's recommendation in Calculus), but will this be sufficient to get anywhere? (At the "end" of all of this, I'd probably read Emil Artin's Galois Theory, or something more modern on the subject.) Is there other mathematics that I'll need but don't have right now (like I said, I'm in the early portion of a vector calc/linear algebra course, and in high school I just went up to calculus and I read Spivak's this past summer)? Am I overlooking some massive obstacle that will ultimately prevent me from getting anywhere (or that will make this project absolutely useless in some way)?
Note that I'm not intending to fully "rediscover" this stuff. I know that it'd be pretty presumptuous to even imagine that I could make the jumps that Galois did. I just want to build a sort of scaffolding for the real thing.
This ended up being way longer than I intended, so, if you got this far (even if you don't respond), thanks for taking time to read all of it.
 A: First, I want to say that I think this is an excellent question (both in content and in the manner it was posed). I'm looking forward to reading others' answers, but here's mine.
In general, I don't really feel there is a lot to be gained by learning mathematics in this way. As Kevin Carlson said in the comments, you first have to overcome the differences in notation and language, and this is nontrivial. This approach will be time-consuming, and the gift of hindsight offers a lot of extra insight that will better serve you in the long run.
HOWEVER, because you asked specifically about Galois Theory, I think there's a great book for you. David Cox wrote a book (http://books.google.com/books/about/Galois_Theory.html?id=3u4RF8SrRooC) which beautifully marries the modern and historical perspectives. I really encourage you to take a look at this book, because I think it is exactly what you're looking for.
A: I have the perfect book for you. It's called Galois' Theory of Algebraic Equations. 
http://www.amazon.com/Galois-Theory-Of-Algebraic-Equations/dp/9810245416
What's really great about it is that it's serious math ... in other words it's not just a historical overview of who did what when; but rather a detailed examination of the math involved in each historical step on the way from antiquity to Abel and Galois.
It's exactly what you're looking for.
Relevant quote from the Amazon blurb:
The main emphasis is placed on equations of at least the third degree, i.e. on the developments during the period from the sixteenth to the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved ...
A: I think there is something to be said for your proposed approach, but also certainly quite a lot to be said against it.
One can read Galois's works --- I actually have a copy sitting on one of the desks in my office, and have read parts of it.  But it is not at all easy going, and it is a misconception to think that, because Galois didn't have all our modern abstract algebra notions available, that you don't need to know them to read his work.
In fact, Galois essentially invented them all from whole cloth, wrote about them in
a very condensed form, and had incredible, far-reaching intuition about them, only a fragment of which he makes explicit, and much of which is left "between the lines".
What this will mean --- I think --- is that you will find Galois incredibly heavy going if you don't already have available all the contemporary interpretations of his work to provide an intellectual framework in which to understand what he is getting at.  (Speaking from my attempts to read his writing, even when you do have
this available, it is often hard to understand what he is getting at --- he is packing a huge number of ideas into a small space, and since there isn't an already existing language that he can use to express them, his writing can be quite oblique at times.)

Perhaps it won't suit you temperamentally, but I think a more sensible thing
to do would be to take a more modern text, but perhaps one that still has something
of a historical focus, and try to learn Galois theory from that instead.
There is also the book The genesis of the abstract group concept by H. Wussing, which gives an extensive explanation of the ideas underlying Galois's
various writings, which you could look at in tandem with a more standard instructional text.

By the way, to say that Galois was "messing around with permutation groups" is 
to massively understate Galois's genius, and also to massively underestimate the difficulty of reading his mathematics.  A better (although still misleadingly
simplistic) mental picture would be to think that all the material on group
theory in Dummitt and Foote was known and obvious to him (probably not quite
true, since as far as I know he didn't know the Sylow theorems in general --- but
then he knew much that isn't in Dummit and Foote).   
A: Even if the abstract group theory was formulated after Galois, there was a lot of studies of concrete groups even before Galois and a lot of theorems in modern algebra was discovered before the algebra became abstract. 
Any group is isomorphic to a subgroup of all one-to-one functions $A\rightarrow A$ for some set $A$, that is, to a subset $\mathcal S$ of those functions, such that $f,g\in\mathcal S\Rightarrow f^{-1},f\circ g\in\mathcal S$. Those permutation elements and their laws was well known to Galois and other mathematicians. If you want to walk in the footsteps of Galois, I guess you have to understand what he known.
There is a theorem saying that for any polynomial over a field (e.g. the field of rational numbers), there is an extended field in which the polynomial has a root. A field extension is a vector space over its ground field, and it turns out that group of automorphisms on the field extension that leaves the ground field invariant has the same number of elements as the dimension of the vector space. I think this was known to mathematicians at the time of Galois.
For a polynomial with roots that are possible to express by repeated root extractions, the groups involved has certain properties.
