These notes are based off those written by Jens Funke for a previous iteration of this course. Those in turn are based on multiple sources, especially [1] and [4]. Here is a brief rundown of a few references you could look at:
We rely a lot on “Fulton and Harris”: [1]. This book takes the point of view that examples should come before theory. It has lots of good exercises. It would be particularly useful for the -theory. It doesn’t discuss Lie groups (as opposed to algebras) much.
Kosmann–Schwarzbach’s book [4] is a relative short source, and is good for its concrete discussion of representations Lie groups.
Serre’s book [5] is unusual in that it develops the theory of finite group representation theory in three chapters at three different levels: the first part, aimed at chemistry students (!) covers character theory and is very concrete; the second takes a more abstract point of view and proves the main theoretical results, while the third deals with issues that come up when you look at representations over fields over than .
James and Liebeck, [3], is an elementary and accessible book on finite group representation theory, with a focus on character theory.
I like Hall, [2], for the theory of linear Lie groups, Lie algebras and representations. It is complete yet readable.