What's a good place to learn Lie groups?

Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to do with groups" So I picked up a copy of Dummit and Foote's book on abstract algebra and skimmed through it. It didn't say anything about Lie groups however. $E_8$ is coming to be rather famous so maybe other people are interested in this question too. Lets suppose I wanted to learn about lie groups. What books should I read to be ready to learn about Lie groups and what is a good book that talks about Lie groups. I'm guessing its a combination of group theory (representation theory in specific) and also differential geometry. Is this correct? Thank you very much for your time.

I think a good place to start with Lie groups (if you don't know Differential Geometry like me) is Brian Hall's Book Lie Groups, Lie algebras and Representations. The strength of such a book for me would be that it talks about matrix Lie groups, e.g. $SO(n),U(n),GL_n, Sp_n,SL_n$ and not general Lie groups in terms of abstract manifolds. Furthermore, the Lie algebra is introduced not as an abstract linear space with a bracket but as the set of all matrices $X$ such that $e^{tX}$ lands in the matrix Lie group for all $t$.

I am using this book now for a course and I find it extremely readable. For one, proofs are presented in almost complete detail and it is easy to follow. By this I mean one does not need a lot of prerequisites to understand the material. You should of course have an understanding of linear algebra, as well as know topological concepts like connectedness, compactness and path-connectedness.

In conclusion, I think the main strength of Hall's Book is that it teaches you ideas through lots and lots of examples. For example, an entire chapter (IIRC chapter 5) is devoted entirely to the representation theory of the Lie algebra $\mathfrak{sl}_3(\Bbb{C})$. I learned a lot from that example there!

You don't need to know any differential geometry to grasp the basic ideas in Lie theory beyond some idea of what a tangent vector is. The study of semisimple Lie groups (which includes $E_8$) is largely algebraic (there are theorems that make this precise but you don't need to know what they are) and getting a good grasp of the important examples doesn't require more than comfort with calculus and linear algebra.

I would recommend Stillwell's Naive Lie Theory in this vein. I agree with Matt E that Fulton and Harris is also a solid resource.

One of the main points of interest with regard to Lie group is their representations, and I think studying them together with their representations makes a lot of sense.

To this end, I recommend Fulton and Harris's book on representation theory. About 3/4 of it is devoted to Lie groups, and it light on the theoretical background (although it does presume some mathematical maturity) and heavy on examples and intuition.

Personally, seeing as you are a high school student, I would start out a little lighter--even lighter than Brian C. Hall's book.

I think the perfect place to get a painless introduction to Lie theory, that gives you the exact right idea without all of the necessary machinery is the little gem of a book "Matrix Groups for Undergraduates" by Tapp. There you will be introduced, in a very congenial and pleasant way, to Lie groups and the ideas of differential geometry simultaneously.

Once you get used to that I would suggest the book by Brian C. Hall that others have mentioned as well as the books by Sepanski and Tom Dieck. In fact, these are the recommended books for the Lie groups part of a course on Lie Groups/Algebraic Groups I'm taking with Jeffrey Adams (one of the big players in the discovery the article you linked to mentioned). You should get a good feel for compact Lie groups before you move onto the more advanced methods needed to discuss non-compact Lie groups.

Also, the notes by Ban and the accompanying lectures are great once you feel prepared to learn about non-compact Lie groups.

Also, an absolutely must read, for when you start learning the more advanced (i.e. anything beyond Tapp's book) topics in Lie groups is the fantastic introductory article Very Basic Lie Theory by Howe.