A good reference for the theorem is the paper by Jean H. Gallier, "What's so special about Kruskal's theorem and the ordinal $\Gamma_0$? A survey of some results in proof theory." Ann. Pure Appl. Logic 53 (1991), no. 3, 199-260.
(With a short erratum in Ann. Pure Appl. Logic 89 (1997), no. 2-3, 275.)
I do not know of any (undergraduate) books where Friedman's result is discussed in any sort of detail, but this paper is very good.
For some background, you may also want to read the paper by Joseph B. Kruskal, "The theory of well-quasi-ordering: A frequently discovered concept." J. Combinatorial Theory Ser. A 13 (1972), 297–305.
Here is the review from MathScinet:
MR0306057 (46 #5184)
This is a survey paper outlining the history and present state of the theory of well-quasi-ordered sets. A wqo is a qo in which each strictly descending sequence is finite and each set of pairwise incomparable elements is finite or, equivalently, each nonempty subset has one but no more than a finite number of nonequivalent minimal elements. If $s$ and $t$ are sequences from a wqo set then $s\leq t$ means that some subsequence of $t$ majorizes $s$ term by term. A basic question is: when is a set of sequences from a wqo set itself wqo? The author traces the history of this problem and points out that over the years many of the results have been rediscovered and republished. This paper and this section in MR should help eliminate further duplication of these results.
Reviewed by P. F. Conrad
A good book to learn about well-quasi-ordering theory itself, from a logician's perspective, is "Recursive Aspects of Descriptive Set Theory" (Oxford Logic Guides), by Richard Mansfield and Galen Weitkamp. I think the level is fairly accessible. The chapter on wqo theory is by S. Simpson, who is a very good expositor.
A primary goal of "Set Theory and Logic" (I put this in quotations because I get the sense you are referring to a particular school of thought, and not just the pure subjects on their own) is to give foundation and motivation to the structures and systems of numbers that we commonly use. As the most basic example, efforts have been made to define natural numbers in terms of sets. Another basic example is the effort to define Mathematics as an extension of Logic.
While these are highly interesting studies, I would classify these types of studies more under the heading of "meta-mathematics" or foundations of mathematics. In essence, this type of study works backwards from the familiar world of numbers and mathematical areas we know, and attempts to ground these structures in well-defined "fundamental" ideas (sorry I have to be vague here, but this stuff is abstract!).
At any rate, from what I've said, you can get the sense that these types of study are not the typical areas a beginner should engage in, unless that beginner be of a more philosophical disposition; in other words, these areas are of a broader nature, and have a different conceptual "flavor". They seek to unify mathematical structures into more basic structures.
On the other hand, the "typical" mathematician works within established fields of math; that is to say, he uses and manipulates the structures and symbols given to him, in an attempt to discover deeper connections and new relationships. He is not usually concerned with foundations, that is a completely seperate study.
So, after I've said all that, my practical advice is to move along the "Algebra -> Pre-Calculus -> Calculus -> etc." route. That gives one the necessary tools for advanced study, and it familiarizes one (at a nice pace!) with what mathematicians really do. And IMHO, Calculus is absolutely essential in this path, because studying that results in a certain understanding and maturity in math that one will need throughout the rest of his mathematical career (e.g., the notions of limit and derivative in Calculus are really fundamental, and are great examples of mathematical intuition and thought).
Just a side note, I am not implying that the independent fields of Logic and Set Theory, as subjects on their own, are "deeply abstract" in the sense I described above (i.e., relate to the foundations of math); but that being said, I do not think they serve as beginning studies either. I believe they fall under the "etc." in the path I mentioned above.
Hope this helps you figure out how you'd like to proceed! Good Luck.
Best Answer
Personally I am taking Keith Devlin's Introduction to Mathematical Thinking on Coursera right now and it is amazing. It changed how I viewed mathematics.
He wrote this really awesome short essay here:
http://spark-public.s3.amazonaws.com/maththink/readings/Background_Reading.pdf
Read that essay. It might change your life. Seriously. The key highlight for me anyway was that mathematics these days is defined as "the science of patterns" and is more about seeing patterns and finding truth. When you phrase it that way it suddenly has much more appeal for people like me who are obsessed with finding patterns by reading history and studying sociology. Also it got me into theoretical CS instead of just hacking random things together.
This pretty much blew me away because I only took up to Linear Algebra in university and I thought math was this stupid calculation based thing.
Proofs are insanely awesome. They're hard and they make you think. And they're beautiful.
https://class.coursera.org/maththink-004
You can sign up for the class here and go through the material at your own pace. I think it's a good idea for someone in your position because it's a survey class. I'm sure you can ask around in the forums for ways to go deeper if you're interested.