Before recommending anything I want to say that to properly understand the general theory for semisimple Lie algebras, it is very useful to treat the sl_2 = SU(2) case (and sl_3 = SU(3) case really) in detail, since sl_2-triples are a very important tool in this case. As such many books include these as specific examples, sometimes entire chapters.
Hall's Lie groups, Lie algebras and representations is good as an elementary introduction. However, it focuses (understandably) on matrix Lie groups which means that the theory is presented in a slightly less general way ar first, although towards the end of the book this is rectified I think. The book does treat sl2 and sl3 quite nicely iirc. Would recommend for a first meeting, but not if you know more.
Duistermaat and Kolk Lie groups is very geometric, too geometric for me, but depends on your background and purpose. It definitely talks about representation theory, but not always in a clear way if you do not already know the concepts. I distinctly remember associated vector bundles being called something different which confused me at first, skill issue though.
Fulton and Harris' Representation theory is a classic, and very good, book on representation theory. Especially if you care about the Lie algebra side of things more it is great (together with Humphrey's, although I havent studied that book much). However, since it deals with quite a lot of general theory, it can be hard to distinguish the essential features of the theory of cpt Lie groups. For example, I don't think the Peter-Weyl theorem is even mentioned? Even though for my applications that is one of the main reasons to care about representations.
Bump's Lie groups is also a good book, but it assumes basic knowledge of representation theory of cpt Lie groups in all but name. This can be supplemented by Hall though. For example, I think Peter-Weyl is proven around page 20 or so lol. So, better if you already know a bit imo.
This post tuned out longer than expected but I hope it is useful!