What's a good place to learn Lie groups?

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.