If you do consult a pure math book on intro to abstract algebra, i recommend the one by mike artin, Algebra. He emphasizes in his "note to the teacher" that he begins his book with matrix operations rather than permutations because "matrix groups are more important". Indeed the first 8 chapters or so of his book discuss mainly matrix groups. Only the chapter called "More group theory" deals with specifically finite groups, and their numerical formulas such as sylow's theorem.
A lie group is a differentiable manifold that is also a group, so in my opinion, as a novice, learning about manifolds for this subject is more important than learning group theory. A good pure mathematics book that introduces manifolds and then lie groups and algebras, is the one by Frank Warner.
Lie groups and lie algebras go hand in hand by the way, since a lie algebra is a vector space, hence a linear approximation to the lie group, precisely in the sense that the tangent space to a manifold approximates the manifold. Indeed the lie algebra of a lie group can be defined as the tangent space of that lie group at the identity, plus some bracket product structure. A common definition of the lie algebra is as the vector space of left - invariant vector fields on the lie group. But since the group structure allows you to uniquely translate any point to any other point, each tangent vector at the origin can be translated everywhere and yields a left invariant vector field. The realization as a vector field let's you define the bracket product as a commutator of the operation of a vector field as a differential operator on the smooth functions.
One simple example is the circle group of unit length complex numbers. The lie algebra is apparently the real line, which is a translate of the tangent line to the circle at the unit element. The fact that the exponential map sending t to e^(it), maps the real line onto the unit circle, has an importamnt generalization to all lie groups. Namely there is always an "exponential map" from the lie algebra to the lie group, at least locally. Take what I say with large grains of salt, as I have never studied this topic. One way to define this exponential map for matrix groups uses the fact that you can plug a matrix into a convergent power series and it will still converge, and the limit will be an invertible matrix.
A more general definition of the exponential map uses the existence theory for differential equations. As I recall from browsing in J. Frank Aadams' book, Lectures on lie groups it goes something like this. Namely starting from a vector at the identity, use the group translation structure to extend to a vector field near the identity. Then solve this "differential equation" to find a curve passing through the identity and having all velocity vectors agreeing with that vector field. Then run along that curve for time t=1, and stop there, and that point is the exponential map evaluated at that original vector.